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Mathematics · Class 12 Science

Ch 4Determinants — Class 12 Mathematics, concept-first.

The study of determinants begins with a question: when does a system of linear equations have a unique solution? For a pair of equations in two variables,

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Determinant Evaluation Using Identities

Expanding a or determinant term by term is painful and error-prone. The smarter route is to transform the determinant into an easy form using properties (the "identities") that change its value in a known, controlled way…

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4.1

Introduction

The study of determinants begins with a question: when does a system of linear equations have a unique solution? For a pair of equations in two variables,

4.2

Determinant

The determinant assigns a single number (real or complex) to every square matrix — formally a function from the set of square matrices to the set of numbers , written .

4.2.1

Determinant of a Matrix of Order One

The simplest case of a determinant arises when the matrix has only one element. For a matrix of order , the determinant is defined in a way that is both natural and consistent with the properties deve…

4.2.2

Determinant of a Matrix of Order Two

For a matrix, the determinant is a single number calculated from its four entries. This number has important geometric interpretations (an area scaling factor) and algebraic properties that make it a…

+Worked Examplesi2 questions
  1. Example 1Evaluate $\begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}$.Free
  2. Example 2Evaluate $\begin{vmatrix} x & x+1 \\ x-1 & x \end{vmatrix}$.Preview
4.2.3

Determinant of a Matrix of Order 3 × 3

11 Q

A determinant of order 3 is found by expressing it in terms of second-order () determinants — a process called expansion of a determinant along a row (or a column).

+Worked Examplesi3 questions
  1. Example 3Evaluate the determinant $\Delta = \begin{vmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{vmatrix}$.Free
  2. Example 4Evaluate $\Delta = \begin{vmatrix} 0 & \sin\alpha & -\cos\alpha \\ -\sin\alpha & 0 & \sin\beta \\ \cos\alpha & -\sin\beta & 0 \end{vmatrix}$…Preview
  3. Example 5Find values of $x$ for which $\begin{vmatrix} 3 & x \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$.Preview
+Exercise 4.1i8 questions
  1. Q1Find the value of the following: $\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}$Free
  2. Q2Find the value of the following: (i) $\begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix}$ (ii) $\begin{vm…Free
  3. Q3If $A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$, then show that $|2A| = 4|A|$.Free
  4. Q4If $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}$, then show that $|3A| = 27|A|$.Preview
  5. Q5Evaluate the determinants (i) $\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}$ (ii) $\begin{vmatrix} 3 & -4 & 5 \\ 1…Preview
  6. Q6If $A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}$, find $|A|$.Preview
  7. Q7Find values of $x$, if (i) $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$ (ii) $\begin{vmatr…Preview
  8. Q8If $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}$, then $x$ is equal to (A) 6 (B) $\pm 6$ (…Preview
4.3

Area of a Triangle

7 Q

You already know the area of a triangle with vertices , , and is given by:

+Worked Examplesi2 questions
  1. Example 6Find the area of the triangle whose vertices are $(3, 8)$, $(-4, 2)$ and $(5, 1)$.Free
  2. Example 7Find the equation of the line joining $A(1, 3)$ and $B(0, 0)$ using determinants and find $k$ if $D(k, 0)$ is a point such that area of tria…Preview
+Exercise 4.2i5 questions
  1. Q1Find area of the triangle with vertices at the point given in each of the following : (i) $(1, 0), (6, 0), (4, 3)$ (ii) $(2, 7), (1, 1), (10…Free
  2. Q2Show that points A $(a, b+c)$, B $(b, c+a)$, C $(c, a+b)$ are collinear.Free
  3. Q3Find values of $k$ if area of triangle is 4 sq. units and vertices are (i) $(k, 0), (4, 0), (0, 2)$ (ii) $(-2, 0), (0, 4), (0, k)$Preview
  4. Q4(i) Find equation of line joining $(1, 2)$ and $(3, 6)$ using determinants. (ii) Find equation of line joining $(3, 1)$ and $(9, 3)$ using d…Preview
  5. Q5If area of triangle is 35 sq units with vertices $(2, -6)$, $(5, 4)$ and $(k, 4)$. Then $k$ is (A) 12 (B) $-2$ (C) $-12, -2$ (D) $12, -2$Preview
4.4

Minors and Cofactors

9 Q

The expansion of a determinant can be written compactly using minors and cofactors, which express it as a sum of products of elements with specially defined coefficients.

+Worked Examplesi4 questions
  1. Example 8Find the minor of element $6$ in the determinant $\Delta = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$.Free
  2. Example 9Find minors and cofactors of all the elements of the determinant $\begin{vmatrix} 1 & -2 \\ 4 & 3 \end{vmatrix}$.Free
  3. Example 10Find minors and cofactors of the elements $a_{11}$, $a_{21}$ in the determinant $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21}…Preview
  4. Example 11Find minors and cofactors of the elements of the determinant $\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$ and verif…Preview
+Exercise 4.3i5 questions
  1. Q1Find the value of the following: (i) $\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}$ (ii) $\begin{vmatrix} a & c \\ b & d \end{vmatrix}$Free
  2. Q2Find the value of the following: (i) $\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}$ (ii) $\begin{vmatrix} 1 & 0 & 4 \\…Free
  3. Q3Using Cofactors of elements of second row, evaluate $\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$.Preview
  4. Q4Using Cofactors of elements of third column, evaluate $\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}$.Preview
  5. Q5If $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$ and $A_{ij}$ is…Preview
4.5

Adjoint and Inverse of a Matrix

The inverse of a matrix was introduced in the previous chapter. Here we establish exactly when an inverse exists and how to find it, using a special matrix called the adjoint of the original matrix.

4.5.1

Adjoint of a Matrix

22 Q

The adjoint of a matrix is the stepping stone to finding its inverse. It is built directly from the cofactors you have already learned to compute.

+Worked Examplesi4 questions
  1. Example 12Find $\operatorname{adj} A$ for $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$.Free
  2. Example 13If $A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$, then verify that $A\,\operatorname{adj} A = |A|\,I$. Also find $…Free
  3. Example 14If $A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix}$, then verify that $(AB)^{-1}…Preview
  4. Example 15Show that the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ satisfies the equation $A^2 - 4A + I = O$, where $I$ is $2 \times 2$…Preview
+Exercise 4.4i18 questions
  1. Q1Find the value of the following: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$Free
  2. Q2$\begin{bmatrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{bmatrix}$ Verify $A (\text{adj } A) = (\text{adj } A) A = |A| I$ in Exercises 3…Free
  3. Q3Find the value of the following: $\begin{bmatrix} 2 & 3 \\ -4 & -6 \end{bmatrix}$Free
  4. Q4$\begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}$ Find the inverse of each of the matrices (if it exists) given in Exerc…Preview
  5. Q5Find the value of the following: $\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}$Preview
  6. Q6Find the value of the following: $\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}$Preview
  7. Q7Find the value of the following: $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$Preview
  8. Q8Find the inverse of the following matrix, if it exists: $A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}$Preview
  9. Q9Find the inverse of the following matrix, if it exists: $A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{bmatrix}$Preview
  10. Q10Find the inverse of the following matrix, if it exists: $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$Preview
  11. Q11Find the value of the following: $\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha & \sin \alpha \\ 0 & \sin \alpha & -\cos \alpha \end{bmatrix}…Preview
  12. Q12Let $A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}$. Verify that $(AB)^{-1} = B^{-…Preview
  13. Q13If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 - 5A + 7I = O$. Hence find $A^{-1}$.Preview
  14. Q14For the matrix $A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$, find the numbers $a$ and $b$ such that $A^2 + aA + bI = O$.Preview
  15. Q15For the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix}$. Show that $A^3 - 6A^2 + 5A + 11 I = O$. Hence, fin…Preview
  16. Q16If $A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$. Verify that $A^3 - 6A^2 + 9A - 4I = O$ and hence find $A^{-1…Preview
  17. Q17Let A be a nonsingular square matrix of order $3 \times 3$. Then $|\text{adj } A|$ is equal to (A) $|A|$ (B) $|A|^2$ (C) $|A|^3$ (D) $3|A|$Preview
  18. Q18If $A$ is an invertible matrix of order 2, then $\det(A^{-1})$ is equal to (A) $\det(A)$ (B) $\dfrac{1}{\det(A)}$ (C) $1$ (D) $0$Preview
4.6

Applications of Determinants and Matrices

Determinants and matrices are powerful tools for solving systems of linear equations. In this section, we focus on systems with two or three variables, using these tools to determine whether a solutio…

4.6.1

Solution of System of Linear Equations Using Inverse of a Matrix

19 Q

Consider a system of three linear equations in three variables , , :

+Worked Examplesi3 questions
  1. Example 16Solve the system of equations $2x + 5y = 1$, $3x + 2y = 7$.Free
  2. Example 17Solve the following system of equations by matrix method. $3x - 2y + 3z = 8$, $2x + y - z = 1$, $4x - 3y + 2z = 4$.Preview
  3. Example 18The sum of three numbers is 6. If we multiply third number by 3 and add second number to it, we get 11. By adding first and third numbers, w…Preview
+Exercise 4.5i16 questions
  1. Q1Examine the consistency of the following system of equations: $x + 2y = 2$ $2x + 3y = 3$Free
  2. Q2Examine the consistency of the following system of equations: $2x - y = 5$ $x + y = 4$Free
  3. Q3Examine the consistency of the following system of equations: $x + 3y = 5$ $2x + 6y = 8$Free
  4. Q4Examine the consistency of the following system of equations: $x + y + z = 1$ $2x + 3y + 2z = 2$ $ax + ay + 2az = 4$Preview
  5. Q5Examine the consistency of the following system of equations: $3x - y - 2z = 2$ $2y - z = -1$ $3x - 5y = 3$Preview
  6. Q6Examine the consistency of the following system of equations: $5x - y + 4z = 5$ $2x + 3y + 5z = 2$ $5x - 2y + 6z = -1$Preview
  7. Q7Solve the following system of linear equations using the matrix method: $5x + 2y = 4$ $7x + 3y = 5$Preview
  8. Q8Solve the following system of linear equations using the matrix method: $2x - y = -2$ $3x + 4y = 3$Preview
  9. Q9Solve the following system of linear equations using the matrix method: $4x - 3y = 3$ $3x - 5y = 7$Preview
  10. Q10Solve the following system of linear equations using the matrix method: $5x + 2y = 3$ $3x + 2y = 5$Preview
  11. Q11Solve the following system of linear equations using the matrix method: $2x + y + z = 1$ $x - 2y - z = \dfrac{3}{2}$ $3y - 5z = 9$Preview
  12. Q12Solve the following system of linear equations using the matrix method: $x - y + z = 4$ $2x + y - 3z = 0$ $x + y + z = 2$Preview
  13. Q13Solve the following system of linear equations using the matrix method: $2x + 3y + 3z = 5$ $x - 2y + z = -4$ $3x - y - 2z = 3$Preview
  14. Q14Solve the following system of linear equations using the matrix method: $x - y + 2z = 7$ $3x + 4y - 5z = -5$ $2x - y + 3z = 12$Preview
  15. Q15If $A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$ solve the system of equations $…Preview
  16. Q16The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ₹ 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is ₹ 90. The cost of 6 kg onion 2…Preview

Miscellaneous Examples

+Miscellaneous Examplesi1 question
  1. Example 19Use product $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \en…Preview

Miscellaneous Exercise on Chapter 4

+Miscellaneous Exercisei9 questions
  1. Q1Prove that the determinant $ \begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix} $ is in…Free
  2. Q2Evaluate $ \begin{vmatrix} \cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha \\ -\sin\beta & \cos\beta & 0 \\ \sin\alpha\cos\beta & \s…Free
  3. Q3If $A^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0…Free
  4. Q4Let $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix}$. Verify that (i) $(\text{adj } A)^{-1} = \text{adj } (A^{-1})$ (…Preview
  5. Q5Evaluate $ \begin{vmatrix} x & y & x+y \\ y & x+y & x \\ x+y & x & y \end{vmatrix} $.Preview
  6. Q6Evaluate $\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$.Preview
  7. Q7Solve the system of equations $\dfrac{2}{x} + \dfrac{3}{y} + \dfrac{10}{z} = 4$, $\dfrac{4}{x} - \dfrac{6}{y} + \dfrac{5}{z} = 1$, $\dfrac{6…Preview
  8. Q8If $x, y, z$ are nonzero real numbers, then the inverse of matrix $A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ is…Preview
  9. Q9Let $A = \begin{bmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{bmatrix}$, where $0 \le \theta \le…Preview

Summary

- Determinant of a square matrix: For a matrix , . For a matrix, expand along any row/column using minors and cofactors.

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 58 questions58 questions
  1. Q1Using the properties of determinants, evaluate: $\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}$Free
  2. Q2Using the properties of determinants, evaluate: $\begin{vmatrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{vmatrix}$Free
  3. Q3Using the properties of determinants, evaluate: $\begin{vmatrix} 0 & xy^2 & xz^2 \\ x^2 y & 0 & yz^2 \\ x^2 z & zy^2 & 0 \end{vmatrix}$Free
  4. Q4Using the properties of determinants, evaluate: $\begin{vmatrix} 3x & -x + y & -x + z \\ x - y & 3y & z - y \\ x - z & y - z & 3z \end{vmatr…Preview
  5. Q5Using the properties of determinants, evaluate: $\begin{vmatrix} x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4 \end{vmatrix}$Preview
  6. Q6Using the properties of determinants, evaluate: $\begin{vmatrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{vmat…Preview
  7. Q7Using the properties of determinants, prove that: $\begin{vmatrix} y^2 z^2 & yz & y + z \\ z^2 x^2 & zx & z + x \\ x^2 y^2 & xy & x + y \end…Preview
  8. Q8Using the properties of determinants, prove that: $\begin{vmatrix} y + z & z & y \\ z & z + x & x \\ y & x & x + y \end{vmatrix} = 4xyz$Preview
  9. Q9Using the properties of determinants, prove that: $\begin{vmatrix} a^2 + 2a & 2a + 1 & 1 \\ 2a + 1 & a + 2 & 1 \\ 3 & 3 & 1 \end{vmatrix} =…Preview
  10. Q10If $A + B + C = 0$, then prove that $\begin{vmatrix} 1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1 \end{vmatrix} = 0$Preview
  11. Q11If the co-ordinates of the vertices of an equilateral triangle with sides of length $a$ are $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, then $…Preview
  12. Q12Find the value of $\theta$ satisfying $\begin{vmatrix} 1 & 1 & \sin 3\theta \\ -4 & 3 & \cos 2\theta \\ 7 & -7 & -2 \end{vmatrix} = 0$.Preview
  13. Q13If $\begin{vmatrix} 4 - x & 4 + x & 4 + x \\ 4 + x & 4 - x & 4 + x \\ 4 + x & 4 + x & 4 - x \end{vmatrix} = 0$, then find values of $x$.Preview
  14. Q14If $a_1, a_2, a_3, \ldots, a_r$ are in G.P., then prove that the determinant $\begin{vmatrix} a_{r+1} & a_{r+5} & a_{r+9} \\ a_{r+7} & a_{r+…Preview
  15. Q15Show that the points $(a + 5,\, a - 4)$, $(a - 2,\, a + 3)$ and $(a,\, a)$ do not lie on a straight line for any value of $a$.Preview
  16. Q16Show that the $\triangle ABC$ is an isosceles triangle if the determinant $\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 + \cos A & 1 + \cos B & 1…Preview
  17. Q17Find $A^{-1}$ if $A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}$ and show that $A^{-1} = \dfrac{A^2 - 3I}{2}$.Preview
  18. Q18If $A = \begin{pmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{pmatrix}$, find $A^{-1}$. Using $A^{-1}$, solve the system of linear eq…Preview
  19. Q19Using matrix method, solve the system of equations $3x + 2y - 2z = 3$, $x + 2y + 3z = 6$, $2x - y + z = 2$.Preview
  20. Q20Given $A = \begin{pmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{pmatrix}$, $B = \begin{pmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2…Preview
  21. Q21If $a + b + c \neq 0$ and $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 0$, then prove that $a = b = c$.Preview
  22. Q22Prove that $\begin{vmatrix} bc - a^2 & ca - b^2 & ab - c^2 \\ ca - b^2 & ab - c^2 & bc - a^2 \\ ab - c^2 & bc - a^2 & ca - b^2 \end{vmatrix}…Preview
  23. Q23If $x + y + z = 0$, prove that $\begin{vmatrix} xa & yb & zc \\ yc & za & xb \\ zb & xc & ya \end{vmatrix} = xyz \begin{vmatrix} a & b & c \…Preview
  24. Q24If $\begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix} = \begin{vmatrix} 6 & -2 \\ 7 & 3 \end{vmatrix}$, then value of $x$ is (A) $3$ (B) $\pm 3$…Preview
  25. Q25The value of determinant $\begin{vmatrix} a - b & b + c & a \\ b - c & c + a & b \\ c - a & a + b & c \end{vmatrix}$ is (A) $a^3 + b^3 + c^3…Preview
  26. Q26The area of a triangle with vertices $(-3, 0)$, $(3, 0)$ and $(0, k)$ is $9$ sq. units. The value of $k$ will be (A) $9$ (B) $3$ (C) $-9$ (D…Preview
  27. Q27The determinant $\begin{vmatrix} b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ac & c - a & ab - a^2 \end{vmatrix}$ equa…Preview
  28. Q28The number of distinct real roots of $\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{…Preview
  29. Q29If $A$, $B$ and $C$ are angles of a triangle, then the determinant $\begin{vmatrix} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B &…Preview
  30. Q30Let $f(t) = \begin{vmatrix} \cos t & t & 1 \\ 2 \sin t & t & 2t \\ \sin t & t & t \end{vmatrix}$, then $\displaystyle\lim_{t \to 0} \dfrac{f…Preview
  31. Q31The maximum value of $\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 + \sin\theta & 1 \\ 1 + \cos\theta & 1 & 1 \end{vmatrix}$ is ($\theta$ is…Preview
  32. Q32If $f(x) = \begin{vmatrix} 0 & x - a & x - b \\ x + a & 0 & x - c \\ x + b & x + c & 0 \end{vmatrix}$, then (A) $f(a) = 0$ (B) $f(b) = 0$ (C…Preview
  33. Q33If $A = \begin{pmatrix} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{pmatrix}$, then $A^{-1}$ exists if (A) $\lambda = 2$ (B) $\lambda \n…Preview
  34. Q34If $A$ and $B$ are invertible matrices, then which of the following is not correct? (A) $\operatorname{adj} A = |A| \cdot A^{-1}$ (B) $\det(…Preview
  35. Q35If $x$, $y$, $z$ are all different from zero and $\begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix} = 0$, then v…Preview
  36. Q36The value of the determinant $\begin{vmatrix} x & x + y & x + 2y \\ x + 2y & x & x + y \\ x + y & x + 2y & x \end{vmatrix}$ is (A) $9x^2(x +…Preview
  37. Q37There are two values of $a$ which make the determinant $\Delta = \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86$,…Preview
  38. Q38If $A$ is a matrix of order $3 \times 3$, then $|3A| = $ ________ .Preview
  39. Q39If $A$ is an invertible matrix of order $3 \times 3$, then $|A^{-1}| = $ ________ .Preview
  40. Q40If $x, y, z \in \mathbb{R}$, then the value of determinant $\begin{vmatrix} (2^x + 2^{-x})^2 & (2^x - 2^{-x})^2 & 1 \\ (3^x + 3^{-x})^2 & (3…Preview
  41. Q41If $\cos 2\theta = 0$, then $\begin{vmatrix} 0 & \cos\theta & \sin\theta \\ \cos\theta & \sin\theta & 0 \\ \sin\theta & 0 & \cos\theta \end{…Preview
  42. Q42If $A$ is a matrix of order $3 \times 3$, then $(A^2)^{-1} = $ ________ .Preview
  43. Q43If $A$ is a matrix of order $3 \times 3$, then the number of minors in the determinant of $A$ are ________ .Preview
  44. Q44The sum of the products of elements of any row with the co-factors of corresponding elements is equal to ________ .Preview
  45. 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 ________ .Preview
  46. Q46$\begin{vmatrix} 0 & x - y & x - z \\ y - x & 0 & y - z \\ z - x & z - y & 0 \end{vmatrix} = $ ________ .Preview
  47. Q47If $f(x) = \begin{vmatrix} (1 + x)^{17} & (1 + x)^{19} & (1 + x)^{23} \\ (1 + x)^{23} & (1 + x)^{29} & (1 + x)^{34} \\ (1 + x)^{41} & (1 + x…Preview
  48. Q48$(A^3)^{-1} = (A^{-1})^3$, where $A$ is a square matrix and $|A| \neq 0$.Preview
  49. Q49$(aA)^{-1} = \dfrac{1}{a} A^{-1}$, where $a$ is any real number and $A$ is a square matrix.Preview
  50. Q50$|A^{-1}| \neq |A|^{-1}$, where $A$ is a non-singular matrix.Preview
  51. Q51If $A$ and $B$ are matrices of order $3$ and $|A| = 5$, $|B| = 3$, then $|3AB| = 27 \times 5 \times 3 = 405$.Preview
  52. Q52If the value of a third order determinant is $12$, then the value of the determinant formed by replacing each element by its co-factor will…Preview
  53. Q53$\begin{vmatrix} x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c \end{vmatrix} = 0$, where $a$, $b$, $c$ are in A.P.Preview
  54. Q54$|\operatorname{adj} A| = |A|^2$, where $A$ is a square matrix of order two.Preview
  55. Q55The determinant $\begin{vmatrix} \sin A & \cos A & \sin A + \cos B \\ \sin B & \cos A & \sin B + \cos B \\ \sin C & \cos A & \sin C + \cos B…Preview
  56. Q56If the determinant $\begin{vmatrix} x + a & p + u & l + f \\ y + b & q + v & m + g \\ z + c & r + w & n + h \end{vmatrix}$ splits into exact…Preview
  57. Q57Let $\Delta = \begin{vmatrix} a & p & x \\ b & q & y \\ c & r & z \end{vmatrix} = 16$, then $\Delta_1 = \begin{vmatrix} p + x & a + x & a +…Preview
  58. Q58The maximum value of $\begin{vmatrix} 1 & 1 & 1 \\ 1 & (1 + \sin\theta) & 1 \\ 1 & 1 & (1 + \cos\theta) \end{vmatrix}$ is $\dfrac{1}{2}$.Preview

CBSE Sample Papers

Questions from official CBSE sample papers.

+Show 8 questions8 questions
  1. Q1If 𝐴 is a square matrix of order 4 and |𝑎𝑑𝑗 𝐴| = 27, then 𝐴 (𝑎𝑑𝑗 𝐴) is equal to (A) 3 (B) 9 (C) 3 𝐼 (D) 9 𝐼Preview
  2. Q2Value of the determinant |cos 67𝑜 sin 67𝑜 sin 23𝑜 cos 23𝑜| is (A) 0 (B) 1 2 (C) √3 2 (D) 1Preview
  3. Q3If for a square matrix $A$, $A. (\text{adj}A) = \begin{bmatrix} 2025 & 0 & 0 \\ 0 & 2025 & 0 \\ 0 & 0 & 2025 \end{bmatrix}$, then the value…Preview
  4. Q4If 𝑨 and 𝑩 are non-singular matrices of same order with 𝒅𝒆𝒕(𝑨) = 𝟓, then [𝒅𝒆𝒕(𝑩−𝟏𝑨𝑩)]² is equal to (A) 5 (B) 25 (C) 45 (D) 55Preview
  5. Q5If the points (𝒙𝟏, 𝒚𝟏), (𝒙𝟐, 𝒚𝟐) and (𝒙𝟏 + 𝒙𝟐, 𝒚𝟏 + 𝒚𝟐) are collinear, then 𝒙𝟏𝒚𝟐 is equal to (A) 𝒙𝟐𝒚𝟏 (B) 𝒙𝟏𝒚𝟏 (C) 𝒙𝟐𝒚𝟐 (D) 𝒙𝟏𝒙𝟐Preview
  6. Q6If $A = \begin{bmatrix} 2 & 0 & 0 \\ -1 & 2 & 3 \\ 3 & 3 & 5 \end{bmatrix}$, then find $A(\text{adj } A)$.Preview
  7. Q7Using properties of determinants, prove that $$\begin{vmatrix} 1 & 1 & 1+3x \\ 1+3y & 1 & 1 \\ 1 & 1+3z & 1 \end{vmatrix} = 9(3xyz + xy + yz…Preview
  8. Q8If $A = \begin{pmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{pmatrix}$, find $A^{-1}$. Use it to solve the system of equations $2x -…Preview