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Mathematics · Class 12 Science

Ch 1Applications of Matrices and Determinants — Class 12 Mathematics, concept-first.

This chapter is about using matrices, not just building their algebra. The 19th-century mathematicians Carl Friedrich Gauss, Camille Jordan, Arthur Cayley and William Rowan Hamilton developed matrix theory precisely to investigate the solutions of systems of linear equations -- the workhorse computational problem behin…

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Adjoint and Inverse of a Matrix

For a square matrix of order , the cofactor of is the signed minor (the minor is the determinant left after deleting row and column ).

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Chapter contents

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

1.1

Introduction

This chapter is about using matrices, not just building their algebra. The 19th-century mathematicians Carl Friedrich Gauss, Camille Jordan, Arthur Cayley and William Rowan Hamilton developed matrix t…

1.2

Inverse of a Non-Singular Square Matrix

Recall that a square matrix of order is non-singular if , and singular if . Matrix algebra defines scalar multiplication, addition and multiplication of matrices, but no division -- a matrix is just a…

1.2.1

Adjoint of a Square Matrix

Let be a square matrix of order , with determinant (also written ). Deleting row and column of leaves a sub-matrix of order ; its determinant is the minor of the entry .

1.2.2

Definition of Inverse Matrix of a Square Matrix

Definition 1.2 (inverse of a square matrix). Let be a square matrix of order . If there exists a square matrix of order with then is called an inverse of , written .

1.2.3

Properties of Inverses of Matrices

Collecting the working laws obeyed by inverses (for non-singular square matrices of the same order, and a scalar):

+Exercise 1.1i15 questions
  1. Q1Find the adjoint of the following: (i) $\begin{pmatrix} -3 & 4 \\ 6 & 2\end{pmatrix}$ (ii) $\begin{pmatrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7…Free
  2. Q2Find the inverse (if it exists) of the following: (i) $\begin{pmatrix} -2 & 4 \\ 1 & -3\end{pmatrix}$ (ii) $\begin{pmatrix} 5 & 1 & 1 \\ 1 &…Free
  3. Q3If $F(\alpha)=\begin{pmatrix}\cos\alpha & 0 & \sin\alpha \\ 0 & 1 & 0 \\ -\sin\alpha & 0 & \cos\alpha\end{pmatrix}$, show that $[F(\alpha)]^…Free
  4. Q4If $A=\begin{pmatrix} 5 & 3 \\ -1 & -2\end{pmatrix}$, show that $A^2-3A-7I_2=O_2$. Hence find $A^{-1}$.Preview
  5. Q5If $A=\dfrac19\begin{pmatrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4\end{pmatrix}$, prove that $A^{-1}=A^T$.Preview
  6. Q6If $A=\begin{pmatrix} 8 & -4 \\ -5 & 3\end{pmatrix}$, verify that $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|I_2$.Preview
  7. Q7If $A=\begin{pmatrix} 3 & 2 \\ 7 & 5\end{pmatrix}$ and $B=\begin{pmatrix} -1 & -3 \\ 5 & 2\end{pmatrix}$, verify that $(AB)^{-1}=B^{-1}A^{-1…Preview
  8. Q8If $\operatorname{adj}(A)=\begin{pmatrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2\end{pmatrix}$, find $A$.Preview
  9. Q9If $\operatorname{adj}(A)=\begin{pmatrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6\end{pmatrix}$, find $A^{-1}$.Preview
  10. Q10Find $\operatorname{adj}(\operatorname{adj}(A))$ if $\operatorname{adj}A=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ -1 & 0 & 1\end{pmatrix}$.Preview
  11. Q11$A=\begin{pmatrix} 1 & \tan x \\ -\tan x & 1\end{pmatrix}$, show that $A^TA^{-1}=\begin{pmatrix}\cos2x & -\sin2x \\ \sin2x & \cos2x\end{pmat…Preview
  12. Q12Find the matrix $A$ for which $A\begin{pmatrix} 5 & 3 \\ -1 & -2\end{pmatrix}=\begin{pmatrix} 14 & 7 \\ 7 & 7\end{pmatrix}$.Preview
  13. Q13Given $A=\begin{pmatrix} 1 & -1 \\ 2 & 0\end{pmatrix}$, $B=\begin{pmatrix} 3 & -2 \\ 1 & 1\end{pmatrix}$ and $C=\begin{pmatrix} 1 & 1 \\ 2 &…Preview
  14. Q14If $A=\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{pmatrix}$, show that $A^{-1}=\dfrac12\left(A^2-3I\right)$.Preview
  15. Q15Decrypt the received encoded message $[2\ {-3}]\,[20\ 4]$ with the encryption matrix $\begin{pmatrix} -1 & -1 \\ 2 & 1\end{pmatrix}$ and the…Preview
1.3

Elementary Transformations of a Matrix

A matrix can be transformed into a related matrix by three permitted operations, called elementary row operations (or, applied to columns, elementary column operations).

1.3.1

Elementary Row and Column Operations

Elementary row (column) operations on a matrix are exactly three moves:

1.3.2

Row-Echelon Form

Using elementary row operations, any non-zero matrix can be reduced to a simplified row-echelon form.

1.3.3

Rank of a Matrix

To define rank precisely, recall: a matrix formed by deleting some rows and some columns of is a sub-matrix of (a matrix is trivially a sub-matrix of itself, deleting zero rows and zero columns), and…

1.3.4

Gauss-Jordan Method

Definition 1.7 (elementary matrix). An elementary matrix is a matrix obtained from the identity matrix by applying exactly one elementary row operation.

+Exercise 1.2i3 questions
  1. Q1Find the rank of the following matrices by minor method: (i) $\begin{pmatrix} 2 & -4 \\ -1 & 2\end{pmatrix}$ (ii) $\begin{pmatrix} -1 & 3 \\…Free
  2. Q2Find the rank of the following matrices by row reduction method: (i) $\begin{pmatrix} 1 & 1 & 1 & 3 \\ 2 & -1 & 3 & 4 \\ 5 & -1 & 7 & 11\end…Preview
  3. Q3Find the inverse of each of the following by Gauss-Jordan method: (i) $\begin{pmatrix} 2 & -1 \\ 5 & -2\end{pmatrix}$ (ii) $\begin{pmatrix}…Preview
1.4

Applications of Matrices: Solving System of Linear Equations

Solving a system of linear equations is one of the most important applications of matrices and determinants.

1.4.1

Formation of a System of Linear Equations

A linear system can be built directly from a real situation. Suppose three shoppers, A, B and C, buy the same brands of rice and sugar in a supermarket: A buys 5 kg rice + 3 kg sugar for Rs.,440; B bu…

1.4.2

System of Linear Equations in Matrix Form

A system of linear equations in unknowns, can be packaged as a single matrix equation. Let be the coefficient matrix (row = the coefficients of equation , in the fixed unknown order ), the column of u…

1.4.3

Solution to a System of Linear Equations

What counts as a solution. For : solving gives the unique pair , which satisfies both equations -- the two lines meet at exactly one point, and the system is called consistent with a unique solution.

1.4.3.1

Matrix Inversion Method

Matrix inversion method applies when the coefficient matrix of is square and non-singular.

+Exercise 1.3i5 questions
  1. Q1Solve the following system of linear equations by matrix inversion method: (i) $2x+5y=-2,\ x+2y=-3$ (ii) $2x-y=8,\ 3x+2y=-2$ (iii) $2x+3y-z=…Free
  2. Q2If $A=\begin{pmatrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1\end{pmatrix}$ and $B=\begin{pmatrix} 1 & 1 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3\end{pm…Free
  3. Q3A man is appointed in a job with a monthly salary of certain amount and a fixed amount of annual increment. If his salary was Rs.\,19{,}800…Preview
  4. Q4Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find t…Preview
  5. Q5The prices of three commodities $A, B$ and $C$ are Rs.\,$x$, $y$ and $z$ per unit respectively. A person $P$ purchases 4 units of $B$ and se…Preview
1.4.3.2

Cramer's Rule

Cramer's rule applies when the coefficient matrix of is square and non-singular; for three equations in three unknowns, let . Then where is with column replaced by the constants (columns unchanged).

+Exercise 1.4i5 questions
  1. Q1Solve the following systems of linear equations by Cramer's rule: (i) $5x-2y+16=0,\ x+3y-7=0$ (ii) $\dfrac3x+2y=12,\ \dfrac2x+3y=13$ (iii) $…Free
  2. Q2In a competitive examination, one mark is awarded for every correct answer while $\dfrac14$ mark is deducted for every wrong answer. A stude…Free
  3. Q3A chemist has one solution which is 50\% acid and another solution which is 25\% acid. How much of each should be mixed to make 10 litres of…Preview
  4. Q4A fish tank can be filled in 10 minutes using both pumps A and B simultaneously. However, pump B can pump water in or out at the same rate.…Preview
  5. Q5A family of 3 people went out for dinner in a restaurant. The cost of two dosai, three idlies and two vadais is Rs.\,150. The cost of two do…Preview
1.4.3.3

Gaussian Elimination Method

Gaussian elimination method row-reduces the augmented matrix to row-echelon form, then reads the unknowns off from the bottom equation upward -- the method of back substitution.

+Exercise 1.5i4 questions
  1. Q1Solve the following systems of linear equations by Gaussian elimination method: (i) $2x-2y+3z=2,\ x+2y-z=3,\ 3x-y+2z=1$ (ii) $2x+4y+6z=22,\…Free
  2. Q2If $ax^2+bx+c$ is divided by $x+3$, $x-5$, and $x-1$, the remainders are $21$, $61$ and $9$ respectively. Find $a, b$ and $c$. (Use Gaussian…Free
  3. Q3An amount of Rs.\,65{,}000 is invested in three bonds at the rates of 6\%, 8\% and 9\% per annum respectively. The total annual income is Rs…Preview
  4. Q4A boy is walking along the path $y=ax^2+bx+c$ through the points $(-6,8), (-2,-12)$, and $(3,8)$. He wants to meet his friend at $P(7,60)$.…Preview
1.5

Applications of Matrices: Consistency of System of Linear Equations by Rank Method

The three methods so far (matrix inversion, Cramer's rule, and to a lesser extent Gaussian elimination applied only to a unique-solution case) leave one question unanswered in general: given any syste…

1.5.1

Non-Homogeneous Linear Equations

Applying the Rouché-Capelli theorem (§1.5) to non-homogeneous systems, four representative cases illustrate every possible outcome:

+Exercise 1.6i3 questions
  1. Q1Test for consistency and if possible, solve the following systems of equations by rank method. (i) $x-y+2z=2,\ 2x+y+4z=7,\ 4x-y+z=4$ (ii) $3…Free
  2. Q2Find the value of $k$ for which the equations $kx-2y+z=1,\ x-2ky+z=-2,\ x-2y+kz=1$ have (i) no solution (ii) unique solution (iii) infinitel…Preview
  3. Q3Investigate the values of $\lambda$ and $\mu$ so that the system of linear equations $2x+3y+5z=9$, $7x+3y-5z=8$, $2x+3y+\lambda z=\mu$ have…Preview
1.5.2

Homogeneous System of Linear Equations

A system is homogeneous when (every constant is ): . Since (the trivial solution) satisfies automatically, always holds -- a homogeneous system is always consistent; the only real question is whether…

+Exercise 1.7i3 questions
  1. Q1Solve the following system of homogeneous equations. (i) $3x+2y+7z=0,\ 4x-3y-2z=0,\ 5x+9y+23z=0$ (ii) $2x+3y-z=0,\ x-y-2z=0,\ 3x+y+3z=0$Free
  2. Q2Determine the values of $\lambda$ for which the following system of equations $x+y+3z=0,\ 4x+3y+\lambda z=0,\ 2x+y+2z=0$ has (i) a unique so…Preview
  3. Q3By using Gaussian elimination method, balance the chemical reaction equation: $C_2H_6+O_2\rightarrow H_2O+CO_2$Preview
1.6

Objective Type Questions

Exercise 1.8 is the chapter's closing multiple-choice review, in the standard "choose the correct or the most suitable answer from the given four alternatives" format.

+Exercise 1.8i25 questions
  1. Q1If $|\operatorname{adj}(\operatorname{adj}A)|=|A|^9$, then the order of the square matrix $A$ is (1) 3 (2) 4 (3) 2 (4) 5Free
  2. Q2If $A$ is a $3\times3$ non-singular matrix such that $AA^T=A^TA$ and $B=A^{-1}A^T$, then $BB^T=$ (1) $A$ (2) $B$ (3) $I_3$ (4) $B^T$Free
  3. Q3If $A=\begin{pmatrix}3 & 5\\ 1 & 2\end{pmatrix}$, $B=\operatorname{adj}A$ and $C=3A$, then $\dfrac{|\operatorname{adj}B|}{|C|}=$ (1) $\dfrac…Free
  4. Q4If $A\begin{pmatrix}1 & -2\\ 1 & 4\end{pmatrix}=\begin{pmatrix}6 & 0\\ 0 & 6\end{pmatrix}$, then $A=$ (1) $\begin{pmatrix}1 & -2\\ 1 & 4\end…Preview
  5. Q5If $A=\begin{pmatrix}7 & 3\\ 4 & 2\end{pmatrix}$, then $9I_2-A=$ (1) $A^{-1}$ (2) $\dfrac{A^{-1}}{2}$ (3) $3A^{-1}$ (4) $2A^{-1}$Preview
  6. Q6If $A=\begin{pmatrix}2 & 0\\ 1 & 5\end{pmatrix}$ and $B=\begin{pmatrix}1 & 4\\ 2 & 0\end{pmatrix}$ then $|\operatorname{adj}(AB)|=$ (1) $-40…Preview
  7. Q7If $P=\begin{pmatrix}1 & x & 0\\ 1 & 3 & 0\\ 2 & 4 & -2\end{pmatrix}$ is the adjoint of the $3\times3$ matrix $A$ and $|A|=4$, then $x$ is (…Preview
  8. Q8If $A=\begin{pmatrix}3 & 1 & -1\\ 2 & -2 & 0\\ 1 & 2 & -1\end{pmatrix}$ and $A^{-1}=\begin{pmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22…Preview
  9. Q9If $A, B$ and $C$ are invertible matrices of some order, then which one of the following is not true? (1) $\operatorname{adj}A=|A|A^{-1}$ (2…Preview
  10. Q10If $(AB)^{-1}=\begin{pmatrix}12 & -17\\ -19 & 27\end{pmatrix}$ and $A^{-1}=\begin{pmatrix}1 & -1\\ -2 & 3\end{pmatrix}$, then $B^{-1}=$ (1)…Preview
  11. Q11If $A^TA^{-1}$ is symmetric, then $A^2=$ (1) $A^{-1}$ (2) $(A^T)^2$ (3) $A^T$ (4) $(A^{-1})^2$Preview
  12. Q12If $A$ is a non-singular matrix such that $A^{-1}=\begin{pmatrix}5 & 3\\ -2 & -1\end{pmatrix}$, then $(A^T)^{-1}=$ (1) $\begin{pmatrix}-5 &…Preview
  13. Q13If $A=\begin{pmatrix}\dfrac35 & \dfrac45\\[4pt] x & \dfrac35\end{pmatrix}$ and $A^T=A^{-1}$, then the value of $x$ is (1) $-\dfrac45$ (2) $-…Preview
  14. Q14If $A=\begin{pmatrix}1 & \tan\dfrac{\theta}{2}\\[4pt] -\tan\dfrac{\theta}{2} & 1\end{pmatrix}$ and $AB=I_2$, then $B=$ (1) $\left(\cos^2\dfr…Preview
  15. Q15If $A=\begin{pmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{pmatrix}$ and $A(\operatorname{adj}A)=\begin{pmatrix}k & 0\\ 0 &…Preview
  16. Q16If $A=\begin{pmatrix}2 & 3\\ 5 & -2\end{pmatrix}$ be such that $\lambda A^{-1}=A$, then $\lambda$ is (1) 17 (2) 14 (3) 19 (4) 21Preview
  17. Q17If $\operatorname{adj}A=\begin{pmatrix}2 & 3\\ 4 & -1\end{pmatrix}$ and $\operatorname{adj}B=\begin{pmatrix}1 & -2\\ -3 & 1\end{pmatrix}$ th…Preview
  18. Q18The rank of the matrix $\begin{pmatrix}1 & 2 & 3 & 4\\ 2 & 4 & 6 & 8\\ -1 & -2 & -3 & -4\end{pmatrix}$ is (1) 1 (2) 2 (3) 4 (4) 3Preview
  19. Q19If $x^ay^b=e^m$, $x^cy^d=e^n$, $\Delta_1=\begin{vmatrix}m & b\\ n & d\end{vmatrix}$, $\Delta_2=\begin{vmatrix}a & m\\ c & n\end{vmatrix}$, $…Preview
  20. Q20Which of the following is/are correct? (i) Adjoint of a symmetric matrix is also a symmetric matrix. (ii) Adjoint of a diagonal matrix is al…Preview
  21. Q21If $\rho(A)=\rho([A|B])$, then the system $AX=B$ of linear equations is (1) consistent and has a unique solution (2) consistent (3) consiste…Preview
  22. Q22If $0\le\theta\le\pi$ and the system of equations $x+(\sin\theta)y-(\cos\theta)z=0$, $(\cos\theta)x-y+z=0$, $(\sin\theta)x+y-z=0$ has a non-…Preview
  23. Q23The augmented matrix of a system of linear equations is $\begin{pmatrix}1 & 2 & 7 & 3\\ 0 & 1 & 4 & 6\\ 0 & 0 & \lambda-7 & \mu+5\end{pmatri…Preview
  24. Q24Let $A=\begin{pmatrix}2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2\end{pmatrix}$ and $4B=\begin{pmatrix}3 & 1 & -1\\ 1 & 3 & x\\ -1 & 1 & 3\end{pma…Preview
  25. Q25If $A=\begin{pmatrix}3 & -3 & 4\\ 2 & -3 & 4\\ 0 & -1 & 1\end{pmatrix}$, then $\operatorname{adj}(\operatorname{adj}A)$ is (1) $\begin{pmatr…Preview
1.7

Summary

Adjoint. transpose of the cofactor matrix of .

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 55 questions55 questions
  1. Q1The system of equations $ax+y+z=0$; $x+by+z=0$; $x+y+cz=0$ has a non-trivial solution then $\dfrac{1}{1-a}+\dfrac{1}{1-b}+\dfrac{1}{1-c}=$ (…Preview
  2. Q2In the homogeneous system $\rho(A) <$ the number of unknowns then the system has : (a) only trivial solution (b) trivial solution and infini…Preview
  3. Q3If A is a matrix of order 3, then $\det(kA)$ is (a) $k^3\det(A)$ (b) $k^2\det(A)$ (c) $k\det(A)$ (d) $\det(A)$Preview
  4. Q4If $A=\begin{bmatrix}2&1\\3&4\end{bmatrix}$, then $(\text{adj }A)A=$ (a) $\begin{bmatrix}\frac15&0\\0&\frac15\end{bmatrix}$ (b) $\begin{bmat…Preview
  5. Q5Solve the system of equations $x+y+2z=4$; $2x+2y+4z=8$; $3x+3y+6z=10$ by using determinant.Preview
  6. Q6Show that the adjoint of $A=\begin{bmatrix}-4&-3&-3\\1&0&1\\4&4&3\end{bmatrix}$ is A itself.Preview
  7. Q7Examine the consistency of the system $x-3y-8z=-10$; $3x+y-4z=0$; $2x+5y+6z-13=0$ by using rank method and hence solve the system.Preview
  8. Q8In echelon form, which of the following is incorrect ? (a) Every row of A which has all its entries 0 occurs below every row which has a non…Preview
  9. Q9If $A = \begin{bmatrix}0 & 0\\0 & 5\end{bmatrix}$, then $A^{12}$ is : (a) $\begin{bmatrix}0 & 0\\0 & 60\end{bmatrix}$ (b) $\begin{bmatrix}0…Preview
  10. Q10If A is a matrix of order 3, then $\det(kA)$ is : (a) $k^3 \det(A)$ (b) $k^2 \det(A)$ (c) $k \det(A)$ (d) $\det(A)$Preview
  11. Q11If $ae^x + be^y = c$; $pe^x + qe^y = d$ and $\Delta_1 = \begin{vmatrix}a & b\\p & q\end{vmatrix}$; $\Delta_2 = \begin{vmatrix}c & b\\d & q\e…Preview
  12. Q12Find the rank of the matrix $\begin{bmatrix}0 & 1 & 2 & 1\\2 & -3 & 0 & -1\\1 & 1 & -1 & 0\end{bmatrix}$.Preview
  13. Q13Find the inverse of the matrix $\begin{bmatrix}3 & 1 & -1\\2 & -2 & 0\\1 & 2 & -1\end{bmatrix}$.Preview
  14. Q14Solve, $x+y+2z=4$, $2x+2y+4z=8$, $3x+3y+6z=12$ by using determinant method.Preview
  15. Q15In a system of 3 linear non-homogeneous equations with three unknowns, if $\Delta = 0$; and $\Delta_x = 0$; $\Delta_y \neq 0$; $\Delta_z = 0…Preview
  16. Q16Which of the following statement is correct regarding homogeneous system ? (a) has only non-trivial solutions (b) always inconsistent (c) ha…Preview
  17. Q17Inverse of $\begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}$ is : (a) $\begin{bmatrix} 3 & -1 \\ -5 & -3 \end{bmatrix}$ (b) $\begin{bmatrix} 2…Preview
  18. Q18If A is a scalar matrix with scalar $k \neq 0$, of order 3, then $A^{-1}$ is : (a) $\dfrac{1}{k}I$ (b) $\dfrac{1}{k^2}I$ (c) $kI$ (d) $\dfra…Preview
  19. Q19Find the adjoint of the matrix $A = \begin{bmatrix}1 & 2\\3 & -5\end{bmatrix}$ and verify the result $A(\text{adj }A) = (\text{adj }A)A = |A…Preview
  20. Q20Solve the following non-homogeneous equations by using determinant method. $2x + 2y + z = 5$ $x - y + z = 1$ $3x + y + 2z = 4$Preview
  21. Q21Discuss the solutions of the following system of equations for all values of $\lambda$. $x + y + z = 2$, $2x + y - 2z = 2$, $\lambda x + y +…Preview
  22. Q22In the homogeneous system $\rho(A)$ is less than the number of unknowns, then the system has : (a) only non-trivial solutions (b) no solutio…Preview
  23. Q23If $A$ is a scalar matrix with scalar $k \neq 0$, of order $3$, then $A^{-1}$ is : (a) $\dfrac{1}{k} I$ (b) $kI$ (c) $\dfrac{1}{k^2} I$ (d)…Preview
  24. Q24To find the number of coins, in each category, write the suitable system of equations for the given situation: "A bag contains 3 types of co…Preview
  25. Q25Prove that $\rho(A) + \rho(B) \ne \rho(A+B)$ by giving the suitable matrices $A$ and $B$ of order 3.Preview
  26. Q26(a) For what values of $\mu$ the system of homogeneous equations $x+y+3z=0$; $4x+3y+\mu z=0$; $2x+y+2z=0$ have: (i) only trivial solution (i…Preview
  27. Q27If $\rho(A) = \rho([A \mid B])$, then the system $AX = B$ of linear equations is : (a) inconsistent (b) consistent and has a unique solution…Preview
  28. Q28If $(AB)^{-1}=\begin{bmatrix}12 & -17\\-19 & 27\end{bmatrix}$ and $A^{-1}=\begin{bmatrix}1 & -1\\-2 & 3\end{bmatrix}$, then $B^{-1}=$ (a) $\…Preview
  29. Q29If $A=\begin{bmatrix}2 & 9\\1 & 7\end{bmatrix}$ then prove that $(A^T)^{-1}=(A^{-1})^T$.Preview
  30. Q30(a) Test the consistency of the following system of linear equations by rank method. $x-y+z=-9$ $2x-y+z=4$ $3x-y+z=6$ $4x-y+2z=7$ **OR** (b)…Preview
  31. Q31If $A=\begin{bmatrix}2 & 3\\5 & -2\end{bmatrix}$ be such that $\lambda A^{-1}=A$, then $\lambda$ is : (a) $19$ (b) $17$ (c) $21$ (d) $14$Preview
  32. Q32Which one of the following is incorrect ? (a) If A is a square matrix of order n, and $\lambda$ is a scalar, then Adj $(\lambda A)=\lambda^n…Preview
  33. Q33Show that the rank of the matrix $\begin{bmatrix}1 & 2 & -1\\3 & -1 & 2\\1 & -2 & 3\\1 & -1 & 1\end{bmatrix}$ is 3.Preview
  34. Q34Solve the following system of linear equations, using matrix inversion method : $5x+2y=3$, $3x+2y=5$.Preview
  35. Q35(a) Cramer's rule is not applicable to solve the system $3x+y+z=2$, $x-3y+2z=1$, $7x-y+4z=5$. Why ? **OR** (b) Prove that the local minimum…Preview
  36. Q36A square matrix A of order n has inverse if and only if : (a) $\rho(A) > n$ (b) $\rho(A) = n$ (c) $\rho(A) \ne n$ (d) $\rho(A) < n$Preview
  37. Q37$|\text{adj}(\text{adj}A)|=|A|^{16}$, then the order of the square matrix A is : (a) $2$ (b) $3$ (c) $5$ (d) $4$Preview
  38. Q38Let $A=\begin{bmatrix}0 & 1\\1 & 1\end{bmatrix}$, $B=\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$ be any two Boolean matrices of the same type.…Preview
  39. Q39Prove that $\begin{bmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}$ is orthogonal.Preview
  40. Q40If $F(\alpha)=\begin{bmatrix}\cos\alpha & 0 & \sin\alpha\\0 & 1 & 0\\-\sin\alpha & 0 & \cos\alpha\end{bmatrix}$, show that $[F(\alpha)]^{-1}…Preview
  41. Q41(a) A boy is walking along the path $y=ax^2+bx+c$ through the points $(-6, 8)$, $(-2, -12)$ and $(3, 8)$. He wants to meet his friend at $P(…Preview
  42. Q42If A is a non-singular matrix such that $A^{-1}=\begin{bmatrix}5 & 3\\-2 & -1\end{bmatrix}$, then $(A^T)^{-1}=$ (a) $\begin{bmatrix}-1 & -3\…Preview
  43. Q43If A, B and C are invertible matrices of some order, then which one of the following is not true ? (a) $\det A^{-1}=(\det A)^{-1}$ (b) $\tex…Preview
  44. Q44Find the rank of the matrix $\begin{bmatrix}-1 & 3\\4 & -7\\3 & -4\end{bmatrix}$.Preview
  45. Q45If $A=\begin{bmatrix}2 & -1 & 3\\-5 & 3 & 1\\-3 & 2 & 3\end{bmatrix}$, then find $|\text{adj}(\text{adj}\,A)|$.Preview
  46. Q46If A is a non-singular matrix of order $3\times3$ and $|A|=5$ then $|A^{-1}|$ is : (a) $5^2$ (b) $5$ (c) $\dfrac{1}{5^2}$ (d) $\dfrac15$Preview
  47. Q47If $A=\begin{bmatrix}2 & 3\\5 & -2\end{bmatrix}$ be such that $\lambda A^{-1}=A$, then $\lambda$ is : (a) $19$ (b) $17$ (c) $21$ (d) $14$Preview
  48. Q48If $\text{adj}\,A=\begin{bmatrix}-1 & 2 & 2\\1 & 1 & 2\\2 & 2 & 1\end{bmatrix}$, find $A^{-1}$.Preview
  49. Q49Solve the system of linear equations $2x+5y=-2$, $x+2y=-3$ by matrix inversion method.Preview
  50. Q50(a) Solve, by Cramer's rule, the system of equations $x_1-x_2=3$, $2x_1+3x_2+4x_3=17$, $x_2+2x_3=7$ **OR** (b) Find the equation of tangent…Preview
  51. Q51If A is a $3\times3$ non-singular matrix such that $AA^T=A^TA$ and $B=A^{-1}A^T$, then $BB^T=$ (a) $I_3$ (b) $A$ (c) $B^T$ (d) $B$Preview
  52. Q52If $\rho(A)=\rho([A\mid B])$, then the system $AX=B$ of linear equations is : (a) consistent and has infinitely many solutions (b) consisten…Preview
  53. Q53If A is a non-singular matrix of odd order, prove that $|\text{adj}\,A|$ is positive.Preview
  54. Q54Find the rank of the matrix $\begin{bmatrix}2 & -2 & 4 & 3\\-3 & 4 & -2 & -1\\6 & 2 & -1 & 7\end{bmatrix}$ by reducing it to an echelon form…Preview
  55. Q55(a) Solve the following system of equations, using matrix inversion method. $2x_1+3x_2+3x_3=5$ $x_1-2x_2+x_3=-4$ $3x_1-x_2-2x_3=3$ **OR** (b…Preview