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Mathematics · Class 11 Science

Ch 4Determinants and Matrices — Class 11 Mathematics, concept-first.

In Class 10 you solved a pair of simultaneous linear equations in two unknowns using a determinant of order two. This chapter extends that idea to determinants of order three, which are the natural tool for solving three simultaneous linear equations in three unknowns, and which show up repeatedly in engineering and ec…

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Determinant of a Matrix

A determinant is a single number computed from a square array of numbers. For order 2, . For order 3, expand along any row or column using cofactors: expanding along row 1, Every determinant can be expanded along any of…

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The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

4.1

Introduction

In Class 10 you solved a pair of simultaneous linear equations in two unknowns using a determinant of order two.

+Exercise 4.1i10 questions
  1. Q1Find the value of determinant $\begin{vmatrix} 2 & -4 \\ 7 & -15 \end{vmatrix}$Free
  2. Q2Find the value of determinant $\begin{vmatrix} 2i & 3 \\ 4 & -i \end{vmatrix}$Free
  3. Q3Find the value of determinant $\begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}$Free
  4. Q4Find the value of determinant $\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}$Preview
  5. Q5Find the value of $x$ if $\begin{vmatrix} x^2-x+1 & x+1 \\ x+1 & x+1 \end{vmatrix} = 0$Preview
  6. Q6Find the value of $x$ if $\begin{vmatrix} x & -1 & 2 \\ 2x & 1 & -3 \\ 3 & -4 & 5 \end{vmatrix} = 29$Preview
  7. Q7Find $x$ and $y$ if $\begin{vmatrix} 4i & i^3 & 2i \\ 1 & 3i^2 & 4 \\ 5 & -3 & i \end{vmatrix} = x+iy$ where $i^2=-1$Preview
  8. Q8Find the minor and cofactor of element of the determinant $D = \begin{vmatrix} 2 & -1 & 3 \\ 1 & 2 & -1 \\ 5 & 7 & 2 \end{vmatrix}$Preview
  9. Q9Evaluate $A = \begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$. Also find minor and cofactor of elements in the $2^{nd}$…Preview
  10. Q10Find the value of determinant expanding along the third column $\begin{vmatrix} -1 & 1 & 2 \\ -2 & 3 & -4 \\ -3 & 4 & 0 \end{vmatrix}$Preview
4.1.1

Value of a Determinant

A determinant of order two is the square arrangement

4.1.2

Determinant of Order 3

Definition. A determinant of order 3 is a square arrangement of 9 elements enclosed between two vertical bars, arranged in 3 rows and 3 columns:

4.1.3

Minors and Cofactors of Elements of a Determinant

Let .

4.2

Properties of Determinants

Expanding a large determinant term-by-term is slow and error-prone. The properties below let us simplify a determinant — using row/column operations — before we ever expand it.

+Exercise 4.2i10 questions
  1. Q11Without expanding evaluate the following determinant $\begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{vmatrix}$Free
  2. Q12Without expanding evaluate the following determinant $\begin{vmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{vmatrix}$Free
  3. Q13Without expanding evaluate the following determinant $\begin{vmatrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{vmatrix}$Free
  4. Q14Prove that $\begin{vmatrix} x+y & y+z & z+x \\ z+x & x+y & y+z \\ y+z & z+x & x+y \end{vmatrix} = 2\begin{vmatrix} x & y & z \\ z & x & y \\…Preview
  5. Q15Using properties of determinant show that $\begin{vmatrix} a+b & a & b \\ a & a+c & c \\ b & c & b+c \end{vmatrix} = 4abc$Preview
  6. Q16Using properties of determinant show that $\begin{vmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \en…Preview
  7. Q17Solve the following equations. $\begin{vmatrix} x+2 & x+6 & x-1 \\ x+6 & x-1 & x+2 \\ x-1 & x+2 & x+6 \end{vmatrix} = 0$Preview
  8. Q18Solve the following equations. $\begin{vmatrix} x-1 & x & x-2 \\ 0 & x-2 & x-3 \\ 0 & 0 & x-3 \end{vmatrix} = 0$Preview
  9. Q19If $\begin{vmatrix} 4+x & 4-x & 4-x \\ 4-x & 4+x & 4-x \\ 4-x & 4-x & 4+x \end{vmatrix} = 0$ then find the values of $x$Preview
  10. Q20Without expanding determinants show that $\begin{vmatrix} 1 & 3 & 6 \\ 6 & 1 & 4 \\ 3 & 7 & 12 \end{vmatrix} + 4\begin{vmatrix} 2 & 3 & 3 \\…Preview
4.3

Applications of Determinants

Having defined determinants and their properties, we now put them to work on three classic problems: solving three simultaneous linear equations in three unknowns (§4.3.1, Cramer's Rule), deciding whe…

+Exercise 4.3i18 questions
  1. Q21Solve the following linear equations by using Cramer's Rule: $x+y+z=6,\ x-y+z=2,\ x+2y-z=2$Free
  2. Q22Solve the following linear equations by using Cramer's Rule: $x+y-2z=-10,\ 2x+y-3z=-19,\ 4x+6y+z=2$Free
  3. Q23Solve the following linear equations by using Cramer's Rule: $x+z=1,\ y+z=1,\ x+y=4$Free
  4. Q24Solve the following linear equations by using Cramer's Rule: $\dfrac{-2}{x}-\dfrac{1}{y}-\dfrac{3}{z}=3,\ \dfrac{2}{x}-\dfrac{3}{y}+\dfrac{1…Preview
  5. Q25The sum of three numbers is 15. If the second number is subtracted from the sum of first and third numbers then we get 5. When the third num…Preview
  6. Q26Examine the consistency of the following equations: $2x-y+3=0,\ 3x+y-2=0,\ 11x+2y-3=0$Preview
  7. Q27Examine the consistency of the following equations: $2x+3y-4=0,\ x+2y=3,\ 3x+4y+5=0$Preview
  8. Q28Examine the consistency of the following equations: $x+2y-3=0,\ 7x+4y-11=0,\ 2x+4y-6=0$Preview
  9. Q29Find k if the following equations are consistent: $2x+3y-2=0,\ 2x+4y-k=0,\ x-2y+3k=0$Preview
  10. Q30Find k if the following equations are consistent: $kx+3y+1=0,\ x+2y+1=0,\ x+y=0$Preview
  11. Q31Find the area of triangle whose vertices are A(5,8), B(5,0), C(1,0)Preview
  12. Q32Find the area of triangle whose vertices are $P\left(\dfrac{3}{2},1\right)$, $Q(4,2)$, $R\left(4,-\dfrac{1}{2}\right)$Preview
  13. Q33Find the area of triangle whose vertices are M(0,5), N(-2,3), T(1,-4)Preview
  14. Q34Find the area of quadrilateral whose vertices are A(-3,1), B(-2,-2), C(3,-1), D(1,4)Preview
  15. Q35Find the value of k, if the area of triangle whose vertices are P(k,0), Q(2,2), R(4,3) is $\dfrac{3}{2}$ sq. unitPreview
  16. Q36Examine the collinearity of the following set of points: A(3,-1), B(0,-3), C(12,5)Preview
  17. Q37Examine the collinearity of the following set of points: P(3,-5), Q(6,1), R(4,2)Preview
  18. Q38Examine the collinearity of the following set of points: $L\left(0,\dfrac{1}{2}\right)$, M(2,-1), $N\left(-4,\dfrac{7}{2}\right)$Preview
4.3.1

Cramer's Rule

Theorem. Consider three linear equations in three variables : where are constants. Provided the (unique) solution is where are obtained from by replacing the column of coefficients of , , respectively…

4.3.2

Consistency of Three Equations in Two Variables

Consider three linear equations in the two variables : These are said to be consistent if they have a common solution.

4.3.3

Area of a Triangle and Collinearity of Three Points

Theorem. If are the vertices of , then its area is

4.3.4

Further Use of Determinants

Determinants reappear throughout the rest of the course. This chapter previews — without developing — six further applications you'll meet later:

4.4

Introduction to Matrices

The theory of matrices was developed by the mathematician Arthur Cayley. Matrices express numerical information compactly and are used to represent operators — they are essential in Economics, Statist…

+Exercise 4.4i28 questions
  1. Q81Construct a matrix $A = [a_{ij}]_{3 \times 2}$ whose elements $a_{ij}$ are given by $a_{ij} = \frac{(i-j)^2}{5-i}$.Free
  2. Q82Construct a matrix $A = [a_{ij}]_{3 \times 2}$ whose elements $a_{ij}$ are given by $a_{ij} = i - 3j$.Free
  3. Q83Construct a matrix $A = [a_{ij}]_{3 \times 2}$ whose elements $a_{ij}$ are given by $a_{ij} = \frac{(i+j)^3}{5}$.Free
  4. Q84Classify the matrix $\begin{bmatrix} 3 & -2 & 4 \\ 0 & 0 & -5 \\ 0 & 0 & 0 \end{bmatrix}$ as a row, a column, a square, a diagonal, a scalar…Preview
  5. Q85Classify the matrix $\begin{bmatrix} 0 & 4 & 7 \\ -4 & 0 & -3 \\ -7 & 3 & 0 \end{bmatrix}$ as a row, a column, a square, a diagonal, a scala…Preview
  6. Q86Classify the matrix $\begin{bmatrix} 5 \\ 4 \\ -3 \end{bmatrix}$ as a row, a column, a square, a diagonal, a scalar, a unit, an upper triang…Preview
  7. Q87Classify the matrix $\begin{bmatrix} 9 & \sqrt{2} & -3 \end{bmatrix}$ as a row, a column, a square, a diagonal, a scalar, a unit, an upper t…Preview
  8. Q88Classify the matrix $\begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}$ as a row, a column, a square, a diagonal, a scalar, a unit, an upper tria…Preview
  9. Q89Classify the matrix $\begin{bmatrix} 2 & 0 & 0 \\ 3 & -1 & 0 \\ -7 & 3 & 1 \end{bmatrix}$ as a row, a column, a square, a diagonal, a scalar…Preview
  10. Q90Classify the matrix $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & \frac{1}{3} \end{bmatrix}$ as a row, a column, a square, a diagonal,…Preview
  11. Q91Classify the matrix $\begin{bmatrix} 10 & -15 & 27 \\ -15 & 0 & \sqrt{34} \\ 27 & \sqrt{34} & \frac{5}{3} \end{bmatrix}$ as a row, a column,…Preview
  12. Q92Classify the matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ as a row, a column, a square, a diagonal, a scalar,…Preview
  13. Q93Classify the matrix $\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ as a row, a column, a square, a diagonal, a scalar,…Preview
  14. Q94Determine whether the following matrix is singular or non-singular: $\begin{bmatrix} a & b & c \\ p & q & r \\ 2a-p & 2b-q & 2c-r \end{bmatr…Preview
  15. Q95Determine whether the following matrix is singular or non-singular: $\begin{bmatrix} 5 & 0 & 5 \\ 1 & 99 & 100 \\ 6 & 99 & 105 \end{bmatrix}…Preview
  16. Q96Determine whether the following matrix is singular or non-singular: $\begin{bmatrix} 3 & 5 & 7 \\ -2 & 1 & 4 \\ 3 & 2 & 5 \end{bmatrix}$Preview
  17. Q97Determine whether the following matrix is singular or non-singular: $\begin{bmatrix} 7 & 5 \\ -4 & 7 \end{bmatrix}$Preview
  18. Q98Find $k$ if the following matrix is singular: $\begin{bmatrix} 7 & 3 \\ -2 & k \end{bmatrix}$Preview
  19. Q99Find $k$ if the following matrix is singular: $\begin{bmatrix} 4 & 3 & 1 \\ 7 & k & 1 \\ 10 & 9 & 1 \end{bmatrix}$Preview
  20. Q100Find $k$ if the following matrix is singular: $\begin{bmatrix} k-1 & 2 & 3 \\ 3 & 1 & 2 \\ 1 & -2 & 4 \end{bmatrix}$Preview
  21. Q101If $A = \begin{bmatrix} -1 & -5 \\ 2 & 0 \\ 3 & -4 \end{bmatrix}$, find $(A^T)^T$.Preview
  22. Q102If $A = \begin{bmatrix} 7 & 3 & 1 \\ -2 & -4 & 1 \\ 5 & 9 & 1 \end{bmatrix}$, find $(A^T)^T$.Preview
  23. Q103Find $a, b, c$ if $\begin{bmatrix} 2 & a & 3 \\ -7 & 4 & 5 \\ c & b & 6 \end{bmatrix}$ is a symmetric matrix.Preview
  24. Q104Find $x, y, z$ if $\begin{bmatrix} 0 & -5i & x \\ y & 0 & z \\ \frac{3}{2} & -\sqrt{2} & 0 \end{bmatrix}$ is a skew symmetric matrix.Preview
  25. Q105For the matrix $\begin{bmatrix} 1 & 2 & -5 \\ 2 & -3 & 4 \\ -5 & 4 & 9 \end{bmatrix}$, using its transpose, state whether it is a symmetric,…Preview
  26. Q106For the matrix $\begin{bmatrix} 2 & 5 & 1 \\ -5 & 4 & 6 \\ -1 & -6 & 3 \end{bmatrix}$, using its transpose, state whether it is a symmetric,…Preview
  27. Q107For the matrix $\begin{bmatrix} 0 & 1+2i & i-2 \\ -1-2i & 0 & -7 \\ 2-i & 7 & 0 \end{bmatrix}$, using its transpose, state whether it is a s…Preview
  28. Q108Construct the matrix $A = [a_{ij}]_{3 \times 3}$ where $a_{ij} = i - j$. State whether $A$ is symmetric or skew symmetric.Preview
4.4.1

Types of Matrices

1. Row matrix. Only one row; order . E.g. , . 2. Column matrix. Only one column; order . E.g. . Note: a single-element matrix like is simultaneously a row and a column matrix. 3. Zero (null) matrix.

4.5

Algebra of Matrices — Equality, Scalar Multiplication and Addition

Four operations make up the algebra of matrices: (1) equality, (2) multiplication by a scalar, (3) addition, (4) multiplication of two matrices (the last one is big enough to need its own section, §4.…

+Exercise 4.5i13 questions
  1. Q109If $A = \begin{bmatrix} 2 & -3 \\ 5 & -4 \\ -6 & 1 \end{bmatrix}$, $B = \begin{bmatrix} -1 & 2 \\ 2 & 2 \\ 0 & 3 \end{bmatrix}$ and $C = \be…Free
  2. Q110If $A = \begin{bmatrix} 2 & -3 \\ 5 & -4 \\ -6 & 1 \end{bmatrix}$, $B = \begin{bmatrix} -1 & 2 \\ 2 & 2 \\ 0 & 3 \end{bmatrix}$ and $C = \be…Free
  3. Q111If $A = \begin{bmatrix} 1 & -2 \\ 5 & 3 \end{bmatrix}$, $B = \begin{bmatrix} 1 & -3 \\ 4 & -7 \end{bmatrix}$, then find the matrix $A - 2B +…Free
  4. Q112If $A = \begin{bmatrix} 1 & 2 & -3 \\ -3 & 7 & -8 \\ 0 & -6 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 9 & -1 & 2 \\ -4 & 2 & 5 \\ 4 & 0 & -3…Preview
  5. Q113If $A = \begin{bmatrix} 1 & -2 \\ 3 & -5 \\ -6 & 0 \end{bmatrix}$, $B = \begin{bmatrix} -1 & -2 \\ 4 & 2 \\ 1 & 5 \end{bmatrix}$ and $C = \b…Preview
  6. Q114Solve the following equations for $X$ and $Y$, if $3X - Y = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$ and $X - 3Y = \begin{bmatrix} 0…Preview
  7. Q115Find matrices $A$ and $B$, if $2A - B = \begin{bmatrix} 6 & -6 & 0 \\ -4 & 2 & 1 \end{bmatrix}$ and $A - 2B = \begin{bmatrix} 3 & 2 & 8 \\ -…Preview
  8. Q116Simplify: $\cos\theta \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} + \sin\theta \begin{bmatrix} \sin\th…Preview
  9. Q117If $A = \begin{bmatrix} i & 2i \\ -3 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2i & i \\ 2 & -3 \end{bmatrix}$, where $\sqrt{-1}=i$, find…Preview
  10. Q118Find $x$ and $y$, if $\begin{bmatrix} 2x+y & -1 & 1 \\ 3 & 4y & 4 \end{bmatrix} + \begin{bmatrix} -1 & 6 & 4 \\ 3 & 0 & 3 \end{bmatrix} = \b…Preview
  11. Q119If $\begin{bmatrix} 2a+b & 3a-b \\ c+2d & 2c-d \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}$, find $a, b, c$ and $d$.Preview
  12. Q120There are two book shops owned by Suresh and Ganesh. Their sales (in Rupees) for books in three subjects — Physics, Chemistry and Mathematic…Preview
  13. Q121Using the same sales matrices $A$ (July) and $B$ (August) as in part (i): if both book shops got 10% profit in the month of August 2017, fin…Preview
4.5.4

Multiplication of Two Matrices

Two matrices and are conformable for the product exactly when the number of columns of equals the number of rows of .

4.6

Properties of Matrix Multiplication

1. Not commutative. In general for matrices (§4.5.4 already showed this). 2. Associative. whenever the orders are suitable for multiplication. 3. Distributive over addition.

+Exercise 4.6i29 questions
  1. Q122Evaluate: $\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} \begin{bmatrix} 2 & -4 & 3 \end{bmatrix}$Free
  2. Q123Evaluate: $\begin{bmatrix} 2 & -1 & 3 \end{bmatrix} \begin{bmatrix} 4 \\ 3 \\ 1 \end{bmatrix}$Free
  3. Q124If $A=\begin{bmatrix}1 & -3\\4 & 2\end{bmatrix}$, $B=\begin{bmatrix}4 & 1\\3 & -2\end{bmatrix}$ show that $AB \neq BA$.Free
  4. Q125If $A=\begin{bmatrix}-1&1&1\\2&3&0\\1&-3&1\end{bmatrix}$, $B=\begin{bmatrix}2&1&4\\3&0&2\\1&2&1\end{bmatrix}$. State whether AB=BA? Justify…Preview
  5. Q126Show that AB=BA where, $A=\begin{bmatrix}-2&3&-1\\-1&2&-1\\-6&9&-4\end{bmatrix}$, $B=\begin{bmatrix}1&3&-1\\2&2&-1\\3&0&-1\end{bmatrix}$Preview
  6. Q127Show that AB=BA where, $A=\begin{bmatrix}\cos\theta & \sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}$, $B=\begin{bmatrix}\cos\phi & -\sin…Preview
  7. Q128If $A=\begin{bmatrix}4&8\\-2&-4\end{bmatrix}$, prove that $A^2=0$.Preview
  8. Q129Verify $A(BC)=(AB)C$ in each of the following cases. $A=\begin{bmatrix}1&0&1\\2&3&0\\0&4&5\end{bmatrix}$, $B=\begin{bmatrix}2&-2\\-1&1\\0&3\…Preview
  9. Q130$A=\begin{bmatrix}2&4&3\\-1&3&2\end{bmatrix}$, $B=\begin{bmatrix}2&-2\\3&3\\-1&1\end{bmatrix}$ and $C=\begin{bmatrix}3&1\\1&3\end{bmatrix}$.Preview
  10. Q131Verify that A(B+C)=AB+BC [printed verbatim; note: the correct left-distributive identity is $A(B+C)=AB+AC$ — see solution] in each of the fo…Preview
  11. Q132$A=\begin{bmatrix}1&-1&3\\2&3&2\end{bmatrix}$, $B=\begin{bmatrix}1&0\\-2&3\\4&3\end{bmatrix}$ and $C=\begin{bmatrix}1&2\\-2&0\\4&-3\end{bmat…Preview
  12. Q133If $A=\begin{bmatrix}1&-2\\5&6\end{bmatrix}$, $B=\begin{bmatrix}3&-1\\3&7\end{bmatrix}$, Find AB-2I, where I is unit matrix of order 2.Preview
  13. Q134If $A=\begin{bmatrix}4&3&2\\-1&2&0\end{bmatrix}$, $B=\begin{bmatrix}1&2\\-1&0\\1&-2\end{bmatrix}$ show that matrix AB is non singular.Preview
  14. Q135If $A=\begin{bmatrix}1&2&0\\5&4&2\\0&7&-3\end{bmatrix}$, find the product $(A+I)(A-I)$.Preview
  15. Q136$A=\begin{bmatrix}\alpha&0\\1&1\end{bmatrix}$, $B=\begin{bmatrix}1&0\\2&1\end{bmatrix}$ find $\alpha$, if $A^2=B$.Preview
  16. Q137If $A=\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}$, Show that $A^2-4A$ is a scalar matrix.Preview
  17. Q138If $A=\begin{bmatrix}1&0\\-1&7\end{bmatrix}$, find k so that $A^2-8A-kI=O$, where I is a unit matrix and O is a null matrix of order 2.Preview
  18. Q139If $A=\begin{bmatrix}8&4\\10&5\end{bmatrix}$, $B=\begin{bmatrix}5&-4\\10&-8\end{bmatrix}$ show that $(A+B)^2=A^2+AB+B^2$.Preview
  19. Q140If $A=\begin{bmatrix}3&1\\-1&2\end{bmatrix}$, prove that $A^2-5A+7I=0$, where I is unit matrix of order 2.Preview
  20. Q141If $A=\begin{bmatrix}3&4\\-4&3\end{bmatrix}$ and $B=\begin{bmatrix}2&1\\-1&2\end{bmatrix}$, show that $(A+B)(A-B)=A^2-B^2$.Preview
  21. Q142If $A=\begin{bmatrix}1&2\\-1&-2\end{bmatrix}$, $B=\begin{bmatrix}2&a\\-1&b\end{bmatrix}$ and if $(A+B)^2=A^2+B^2$. find values of a and b.Preview
  22. Q143Find matrix X such that AX=B, where $A=\begin{bmatrix}1&-2\\-2&1\end{bmatrix}$ and $B=\begin{bmatrix}-3\\-1\end{bmatrix}$.Preview
  23. Q144Find k, if $A=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$ and if $A^2=kA-2I$.Preview
  24. Q145Find x, if $\begin{bmatrix}1 & x & 1\end{bmatrix}\begin{bmatrix}1&2&3\\4&5&6\\3&2&5\end{bmatrix}\begin{bmatrix}1\\-2\\3\end{bmatrix}=0$.Preview
  25. Q146Find x and y, if $\left\{4\begin{bmatrix}2&-1&3\\1&0&2\end{bmatrix}-\begin{bmatrix}3&-3&4\\2&1&1\end{bmatrix}\right\}\begin{bmatrix}2\\-1\\1…Preview
  26. Q147Find x, y, z if $\left\{3\begin{bmatrix}2&0\\0&2\\2&2\end{bmatrix}-4\begin{bmatrix}1&1\\-1&2\\3&1\end{bmatrix}\right\}\begin{bmatrix}1\\2\en…Preview
  27. Q148If $A=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix}$, show that $A^2=\begin{bmatrix}\cos2\alpha&\sin2\alpha\\-\s…Preview
  28. Q149If $A=\begin{bmatrix}1&2\\3&5\end{bmatrix}$, $B=\begin{bmatrix}0&4\\2&-1\end{bmatrix}$, show that $AB \neq BA$, but $|AB|=|A|.|B|$Preview
  29. Q150Jay and Ram are two friends in a class. Jay wanted to buy 4 pens and 8 notebooks, Ram wanted to buy 5 pens and 12 notebooks. Both of them we…Preview
4.7

Properties of Transpose of a Matrix

1. for any matrix . 2. for a constant . 3. , for of the same order. 4. — the transpose of a product reverses the order of the factors. More generally, . 5. If is symmetric, . 6.

+Exercise 4.7i20 questions
  1. Q151Find $A^T$, if $A=\begin{bmatrix}1 & 3\\-4 & 5\end{bmatrix}$Free
  2. Q152Find $A^T$, if $A=\begin{bmatrix}2 & -6 & 1\\-4 & 0 & 5\end{bmatrix}$Free
  3. Q153If $[a_{ij}]_{3\times 3}$ where $a_{ij}=2(i-j)$. Find $A$ and $A^T$. State whether $A$ and $A^T$ are symmetric or skew symmetric matrices?Free
  4. Q154If $A=\begin{bmatrix}5 & -3\\4 & -3\\-2 & 1\end{bmatrix}$, Prove that $(2A)^T=2A^T$.Preview
  5. Q155If $A=\begin{bmatrix}1 & 2 & -5\\2 & -3 & 4\\-5 & 4 & 9\end{bmatrix}$, Prove that $(3A)^T=3A^T$.Preview
  6. Q156If $A=\begin{bmatrix}0 & 1+2i & i-2\\-1-2i & 0 & -7\\2-i & 7 & 0\end{bmatrix}$ where $i=\sqrt{-1}$, Prove that $A^T=-A$.Preview
  7. Q157If $A=\begin{bmatrix}2 & -3\\5 & -4\\-6 & 1\end{bmatrix}$, $B=\begin{bmatrix}2 & 1\\4 & -1\\-3 & 3\end{bmatrix}$ and $C=\begin{bmatrix}1 & 2…Preview
  8. Q158If $A=\begin{bmatrix}2 & -3\\5 & -4\\-6 & 1\end{bmatrix}$, $B=\begin{bmatrix}2 & 1\\4 & -1\\-3 & 3\end{bmatrix}$ and $C=\begin{bmatrix}1 & 2…Preview
  9. Q159If $A=\begin{bmatrix}5 & 4\\-2 & 3\end{bmatrix}$ and $B=\begin{bmatrix}-1 & 3\\4 & -1\end{bmatrix}$, then find $C^T$, such that $3A-2B+C=I$,…Preview
  10. Q160If $A=\begin{bmatrix}7 & 3 & 0\\0 & 4 & -2\end{bmatrix}$, $B=\begin{bmatrix}0 & -2 & 3\\2 & 1 & -4\end{bmatrix}$ then find $A^T+4B^T$Preview
  11. Q161If $A=\begin{bmatrix}7 & 3 & 0\\0 & 4 & -2\end{bmatrix}$, $B=\begin{bmatrix}0 & -2 & 3\\2 & 1 & -4\end{bmatrix}$ then find $5A^T-5B^T$Preview
  12. Q162If $A=\begin{bmatrix}1 & 0 & 1\\3 & 1 & 2\end{bmatrix}$, $B=\begin{bmatrix}2 & 1 & -4\\3 & 5 & -2\end{bmatrix}$ and $C=\begin{bmatrix}0 & 2…Preview
  13. Q163If $A=\begin{bmatrix}-1 & 2 & 1\\-3 & 2 & -3\end{bmatrix}$ and $B=\begin{bmatrix}2 & 1\\-3 & 2\\-1 & 3\end{bmatrix}$, prove that $(A+B^T)^T=…Preview
  14. Q164Prove that $A+A^T$ is a symmetric and $A-A^T$ is a skew symmetric matrix, where $A=\begin{bmatrix}1 & 2 & 4\\3 & 2 & 1\\-2 & -3 & 2\end{bmat…Preview
  15. Q165Prove that $A+A^T$ is a symmetric and $A-A^T$ is a skew symmetric matrix, where $A=\begin{bmatrix}5 & 2 & -4\\3 & -7 & 2\\4 & -5 & -3\end{bm…Preview
  16. Q166Express the following matrix as the sum of a symmetric and a skew symmetric matrix: $\begin{bmatrix}4 & -2\\3 & -5\end{bmatrix}$Preview
  17. Q167Express the following matrix as the sum of a symmetric and a skew symmetric matrix: $\begin{bmatrix}3 & 3 & -1\\-2 & -2 & 1\\-4 & -5 & 2\end…Preview
  18. Q168If $A=\begin{bmatrix}2 & -1\\3 & -2\\4 & 1\end{bmatrix}$ and $B=\begin{bmatrix}0 & 3 & -4\\2 & -1 & 1\end{bmatrix}$, verify that $(AB)^T=B^T…Preview
  19. Q169If $A=\begin{bmatrix}2 & -1\\3 & -2\\4 & 1\end{bmatrix}$ and $B=\begin{bmatrix}0 & 3 & -4\\2 & -1 & 1\end{bmatrix}$, verify that $(BA)^T=A^T…Preview
  20. Q170If $A=\begin{bmatrix}\cos\alpha & \sin\alpha\\-\sin\alpha & \cos\alpha\end{bmatrix}$, show that $A^TA=I$, where $I$ is the unit matrix of or…Preview
4.8

Let's Remember — Chapter Summary

- Order-3 determinant expansion (row 1): . - Minor/Cofactor: = determinant left after deleting row , column ; .

More questions

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+Show 42 questions42 questions
  1. Q39The determinant $D=\begin{vmatrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \end{vmatrix} = 0$ if A) a, b, c are in A.P. B) a, b, c are i…Free
  2. Q40If $\begin{vmatrix} x^k & x^{k+2} & x^{k+3} \\ y^k & y^{k+2} & y^{k+3} \\ z^k & z^{k+2} & z^{k+3} \end{vmatrix} = (x-y)(y-z)(z-x)\left(\dfra…Free
  3. Q41Let $D=\begin{vmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\si…Free
  4. Q42The value of a for which the system of equations $a^3x+(a+1)^3y+(a+2)^3z=0$, $ax+(a+1)y+(a+2)z=0$ and $x+y+z=0$ has a non-zero solution is A…Preview
  5. Q43$\begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix} = $ A) $2\begin{vmatrix} c & b & a \\ r & q & p \\ z &…Preview
  6. Q44The system $3x-y+4z=3$, $x+2y-3z=-2$ and $6x+5y+\lambda z=-3$ has at least one solution when A) $\lambda=-5$ B) $\lambda=5$ C) $\lambda=3$ D…Preview
  7. Q45If $x=-9$ is a root of $\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix} = 0$ then the other two roots are A) 2, -7 B) -2,…Preview
  8. Q46If $\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x+iy$ then A) x=3, y=1 B) x=1, y=3 C) x=0, y=3 D) x=0, y=0Preview
  9. Q47If A(0,0), B(1,3) and C(k,0) are vertices of triangle ABC whose area is 3 sq. units then the value of k is A) 2 B) -3 C) 3 or -3 D) -2 or +2Preview
  10. Q48Which of the following is correct A) Determinant is a square matrix B) Determinant is a number associated to a matrix C) Determinant is a nu…Preview
  11. Q49Evaluate: $\begin{vmatrix} 2 & -5 & 7 \\ 5 & 2 & 1 \\ 9 & 0 & 2 \end{vmatrix}$Preview
  12. Q50Evaluate: $\begin{vmatrix} 1 & -3 & 12 \\ 0 & 2 & -4 \\ 9 & 7 & 2 \end{vmatrix}$Preview
  13. Q51Evaluate the determinant along the second column: $\begin{vmatrix} 1 & -1 & 2 \\ 3 & 2 & -2 \\ 0 & 1 & -2 \end{vmatrix}$Preview
  14. Q52Evaluate: $\begin{vmatrix} 2 & 3 & 5 \\ 400 & 600 & 1000 \\ 48 & 47 & 18 \end{vmatrix}$Preview
  15. Q53Evaluate by using properties: $\begin{vmatrix} 101 & 102 & 103 \\ 106 & 107 & 108 \\ 1 & 2 & 3 \end{vmatrix}$Preview
  16. Q54Find the minor and cofactor of every element of the determinant: $\begin{vmatrix} -1 & 0 & 4 \\ -2 & 1 & 3 \\ 0 & -4 & 2 \end{vmatrix}$Preview
  17. Q55Find the minor and cofactor of every element of the determinant: $\begin{vmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{vmatrix}$Preview
  18. Q56Find the value of x if $\begin{vmatrix} 1 & 4 & 20 \\ 1 & -2 & -5 \\ 1 & 2x & 5x^2 \end{vmatrix} = 0$Preview
  19. Q57Find the value of x if $\begin{vmatrix} 1 & 2x & 4x \\ 1 & 4 & 16 \\ 1 & 1 & 1 \end{vmatrix} = 0$Preview
  20. Q58By using properties of determinant prove that $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix} = 0$Preview
  21. Q59Without expanding the determinant show that $\begin{vmatrix} b+c & bc & b^2c^2 \\ c+a & ca & c^2a^2 \\ a+b & ab & a^2b^2 \end{vmatrix} = 0$Preview
  22. Q60Without expanding the determinant show that $\begin{vmatrix} xa & yb & zc \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1 \end{vmatrix} = \begin{vmatrix} x…Preview
  23. Q61Without expanding the determinant show that $\begin{vmatrix} l & m & n \\ e & d & f \\ u & v & w \end{vmatrix} = \begin{vmatrix} n & f & w \…Preview
  24. Q62Without expanding the determinant show that $\begin{vmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{vmatrix} = 0$Preview
  25. Q63If $\begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix} = 0$ then show that $\dfrac{1}{1-a}+\dfrac{1}{1-b}+\dfrac{1}{1-c}=1$Preview
  26. Q64Solve the following linear equations by Cramer's Rule: $2x-y+z=1,\ x+2y+3z=8,\ 3x+y-4z=1$Preview
  27. Q65Solve the following linear equations by Cramer's Rule: $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{3}{2},\ \dfrac{1}{y}+\dfrac{1}{z}=\dfrac{5}{6},\ \d…Preview
  28. Q66Solve the following linear equations by Cramer's Rule: $2x+3y+3z=5,\ x-2y+z=-4,\ 3x-y-2z=3$Preview
  29. Q67Solve the following linear equations by Cramer's Rule: $x-y+2z=7,\ 3x+4y-5z=5,\ 2x-y+3z=12$Preview
  30. Q68Find the value of k if the following equations are consistent: $(k+1)x+(k-1)y+(k-1)=0,\ (k-1)x+(k+1)y+(k-1)=0,\ (k-1)x+(k-1)y+(k+1)=0$Preview
  31. Q69Find the value of k if the following equations are consistent: $3x+y-2=0,\ kx+2y-3=0,\ 2x-y=3$Preview
  32. Q70Find the value of k if the following equations are consistent: $(k-2)x+(k-1)y=17,\ (k-1)x+(k-2)y=18,\ x+y=5$Preview
  33. Q71Find the area of triangle whose vertices are A(-1,2), B(2,4), C(0,0)Preview
  34. Q72Find the area of triangle whose vertices are P(3,6), Q(-1,3), R(2,-1)Preview
  35. Q73Find the area of triangle whose vertices are L(1,1), M(-2,2), N(5,4)Preview
  36. Q74Find the value of k if the area of triangle is 4 square units and the vertices are P(k,0), Q(4,0), R(0,2)Preview
  37. Q75Find the value of k if the area of triangle is $\dfrac{33}{2}$ square units and the vertices are L(3,-5), M(-2,k), N(1,4)Preview
  38. Q76Find the area of quadrilateral whose vertices are A(0,-4), B(4,0), C(-4,0), D(0,4)Preview
  39. Q77An amount of ₹5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is…Preview
  40. Q78Show that the lines $x-y=6$, $4x-3y=20$ and $6x+5y+8=0$ are concurrent. Also find the point of concurrence.Preview
  41. Q79Show that the following points are collinear by determinant method: L(2,5), M(5,7), N(8,9)Preview
  42. Q80Show that the following points are collinear by determinant method: P(5,1), Q(1,-1), R(11,4)Preview
+Show 42 questions42 questions
  1. Q171Given $A=\begin{bmatrix}1 & 3\\2 & 2\end{bmatrix}$, $I=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ if $A-\lambda I$ is a singular matrix then…Free
  2. Q172Consider the matrices $A=\begin{bmatrix}4 & 6 & -1\\3 & 0 & 2\\1 & -2 & 5\end{bmatrix}$, $B=\begin{bmatrix}2 & 4\\0 & 1\\-1 & 2\end{bmatrix}…Free
  3. Q173If A and B are square matrices of equal order, then which one is correct among the following? (A) $A+B=B+A$ (B) $A+B=A-B$ (C) $A-B=B-A$ (D)…Free
  4. Q174If $A=\begin{bmatrix}1 & 2 & 2\\2 & 1 & -2\\a & 2 & b\end{bmatrix}$ is a matrix satisfying the equation $AA^T=9I$, where $I$ is the identity…Preview
  5. Q175If $A=\begin{bmatrix}\alpha & 2\\2 & \alpha\end{bmatrix}$ and $|A^3|=125$, then $\alpha=$ ....... (A) $\pm 3$ (B) $\pm 2$ (C) $\pm 5$ (D) $0…Preview
  6. Q176If $\begin{bmatrix}5 & 7\\x & 1\\2 & 6\end{bmatrix}-\begin{bmatrix}1 & 2\\-3 & 5\\2 & y\end{bmatrix}=\begin{bmatrix}4 & 5\\4 & -4\\0 & 4\end…Preview
  7. Q177If $A+B=\begin{bmatrix}7 & 4\\8 & 9\end{bmatrix}$ and $A-B=\begin{bmatrix}1 & 2\\0 & 3\end{bmatrix}$ then the value of $A$ is ....... (A) $\…Preview
  8. Q178If $\begin{bmatrix}x & 3x-y\\zx+z & 3y-w\end{bmatrix}=\begin{bmatrix}3 & 2\\4 & 7\end{bmatrix}$ then .......... (A) $x=3,y=7,z=1,w=14$ (B) $…Preview
  9. Q179For suitable matrices A, B, the false statement is ....... (A) $(AB)^T=A^TB^T$ (B) $(A^T)^T=A$ (C) $(A-B)^T=A^T-B^T$ (D) $(A+B)^T=A^T+B^T$Preview
  10. Q180If $A=\begin{bmatrix}-2 & 1\\0 & 3\end{bmatrix}$ and $f(x)=2x^2-3x$, then $f(A)=$ ......... (A) $\begin{bmatrix}14 & 1\\0 & -9\end{bmatrix}$…Preview
  11. Q181If $A=\text{diag}[2\ {-3}\ {-5}]$, $B=\text{diag}[4\ {-6}\ {-3}]$ and $C=\text{diag}[-3\ 4\ 1]$ then find $B+C-A$Preview
  12. Q182If $A=\text{diag}[2\ {-3}\ {-5}]$, $B=\text{diag}[4\ {-6}\ {-3}]$ and $C=\text{diag}[-3\ 4\ 1]$ then find $2A+B-5C$Preview
  13. Q183If $f(\alpha)=A=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0\\\sin\alpha & \cos\alpha & 0\\0 & 0 & 1\end{bmatrix}$, Find $f(-\alpha)$Preview
  14. Q184If $f(\alpha)=A=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0\\\sin\alpha & \cos\alpha & 0\\0 & 0 & 1\end{bmatrix}$, Find $f(-\alpha)+f(\alpha…Preview
  15. Q185Find matrices A and B, where $2A-B=\begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}$ and $A+3B=\begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}$Preview
  16. Q186Find matrices A and B, where $3A-B=\begin{bmatrix}-1 & 2 & 1\\1 & 0 & 5\end{bmatrix}$ and $A+5B=\begin{bmatrix}0 & 0 & 1\\-1 & 0 & 0\end{bma…Preview
  17. Q187If $A=\begin{bmatrix}2 & -3\\3 & -2\\-1 & 4\end{bmatrix}$, $B=\begin{bmatrix}-3 & 4 & 1\\2 & -1 & -3\end{bmatrix}$, Verify $(A+B^T)^T=A^T+2B…Preview
  18. Q188If $A=\begin{bmatrix}2 & -3\\3 & -2\\-1 & 4\end{bmatrix}$, $B=\begin{bmatrix}-3 & 4 & 1\\2 & -1 & -3\end{bmatrix}$, Verify $(3A-5B^T)^T=3A^T…Preview
  19. Q189If $A=\begin{bmatrix}\cos\alpha & -\sin\alpha\\\sin\alpha & \cos\alpha\end{bmatrix}$ and $A+A^T=I$, where $I$ is unit matrix $2\times 2$, th…Preview
  20. Q190If $A=\begin{bmatrix}1 & 2\\3 & 2\\-1 & 0\end{bmatrix}$ and $B=\begin{bmatrix}1 & 3 & 2\\4 & -1 & -3\end{bmatrix}$, show that $AB$ is singul…Preview
  21. Q191If $A=\begin{bmatrix}1 & 2 & 3\\2 & 4 & 6\\1 & 2 & 3\end{bmatrix}$, $B=\begin{bmatrix}1 & -1 & 1\\-3 & 2 & -1\\-2 & 1 & 0\end{bmatrix}$, sho…Preview
  22. Q192If $A=\begin{bmatrix}1 & -1 & 0\\2 & 3 & 4\\0 & 1 & 2\end{bmatrix}$, $B=\begin{bmatrix}2 & 2 & -4\\-4 & 2 & -4\\2 & -1 & 5\end{bmatrix}$, sh…Preview
  23. Q193If $A=\begin{bmatrix}2 & 1\\0 & 3\end{bmatrix}$, $B=\begin{bmatrix}1 & 2\\3 & -2\end{bmatrix}$, verify that $|AB|=|A||B|$.Preview
  24. Q194If $A_\alpha=\begin{bmatrix}\cos\alpha & \sin\alpha\\-\sin\alpha & \cos\alpha\end{bmatrix}$, show that $A_\alpha\cdot A_\beta=A_{\alpha+\bet…Preview
  25. Q195If $A=\begin{bmatrix}1 & \omega\\\omega^2 & 1\end{bmatrix}$, $B=\begin{bmatrix}\omega^2 & 1\\1 & \omega\end{bmatrix}$, where $\omega$ is a c…Preview
  26. Q196If $A=\begin{bmatrix}2 & -2 & -4\\-1 & 3 & 4\\1 & -2 & -3\end{bmatrix}$ show that $A^2=A$.Preview
  27. Q197If $A=\begin{bmatrix}4 & -1 & -4\\3 & 0 & -4\\3 & -1 & -3\end{bmatrix}$, show that $A^2=I$.Preview
  28. Q198If $A=\begin{bmatrix}3 & -5\\-4 & 2\end{bmatrix}$, show that $A^2-5A-14I=0$.Preview
  29. Q199If $A=\begin{bmatrix}2 & -1\\-1 & 2\end{bmatrix}$, show that $A^2-4A+3I=0$.Preview
  30. Q200If $A=\begin{bmatrix}-3 & 2\\2 & -4\end{bmatrix}$, $B=\begin{bmatrix}1 & x\\y & 0\end{bmatrix}$, and $(A+B)(A-B)=A^2-B^2$, find $x$ and $y$.Preview
  31. Q201If $A=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$ and $B=\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}$ show that $(A+B)(A-B)\neq A^2-B^2$.Preview
  32. Q202If $A=\begin{bmatrix}2 & -1\\3 & -2\end{bmatrix}$, find $A^3$.Preview
  33. Q203Find $x,y$ if $\begin{bmatrix}0 & -1 & 4\end{bmatrix}\left\{2\begin{bmatrix}4 & 5\\3 & 6\\2 & -1\end{bmatrix}+3\begin{bmatrix}4 & 3\\1 & 4\\…Preview
  34. Q204Find $x,y$ if $\left\{-1\begin{bmatrix}1 & 2 & 1\\2 & 0 & 3\end{bmatrix}+3\begin{bmatrix}2 & -3 & 7\\1 & -1 & 3\end{bmatrix}\right\}\begin{b…Preview
  35. Q205Find $x,y,z$ if $\left\{5\begin{bmatrix}0 & 1\\1 & 0\\1 & 1\end{bmatrix}-3\begin{bmatrix}2 & 1\\3 & -2\\1 & 3\end{bmatrix}\right\}\begin{bma…Preview
  36. Q206Find $x,y,z$ if $\left\{\begin{bmatrix}1 & 3 & 2\\2 & 0 & 1\\3 & 1 & 2\end{bmatrix}+2\begin{bmatrix}3 & 0 & 2\\1 & 4 & 5\\2 & 1 & 0\end{bmat…Preview
  37. Q207If $A=\begin{bmatrix}2 & 1 & -3\\0 & 2 & 6\end{bmatrix}$, $B=\begin{bmatrix}1 & 0 & -2\\3 & -1 & 4\end{bmatrix}$, find $AB^T$ and $A^TB$.Preview
  38. Q208If $A=\begin{bmatrix}2 & -4\\3 & -2\\0 & 1\end{bmatrix}$, $B=\begin{bmatrix}1 & -1 & 2\\-2 & 1 & 0\end{bmatrix}$, show that $(AB)^T=B^TA^T$.Preview
  39. Q209If $A=\begin{bmatrix}3 & -4\\1 & -1\end{bmatrix}$, prove that $A^n=\begin{bmatrix}1+2n & -4n\\n & 1-2n\end{bmatrix}$, for all $n\in N$.Preview
  40. Q210If $A=\begin{bmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}$, prove that $A^n=\begin{bmatrix}\cos n\theta & -\sin n\…Preview
  41. Q211Two farmers Shantaram and Kantaram cultivate three crops rice, wheat and groundnut. The sale (In Rupees) of these crops by both the farmers…Preview
  42. Q212Two farmers Shantaram and Kantaram cultivate three crops rice, wheat and groundnut. The sale (In Rupees) of these crops by both the farmers…Preview