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Mathematics · Class 11 Science

Ch 7Matrices and Determinants — Class 11 Mathematics, concept-first.

The beginnings of matrices and determinants go back to the second century BC, with traces as far back as the fourth century BC — though it was only near the end of the seventeenth century that these ideas resurfaced and their real development began.

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12

Concepts

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Key concepts

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Types of Matrices

A matrix is a rectangular array of numbers (or real-valued functions) arranged in rows and columns, enclosed in square brackets.

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Chapter contents

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Introduction

The beginnings of matrices and determinants go back to the second century BC, with traces as far back as the fourth century BC — though it was only near the end of the seventeenth century that these i…

7.2

Matrices

A matrix is a rectangular array of entries — arranged in rows and columns and enclosed in square brackets — usually named with capital letters .

7.2.1

Types of Matrices

Row and column matrices. A matrix with only one row (order ) is a row matrix, e.g. . A matrix with only one column (order ) is a column matrix.

7.2.2

Equality of Matrices

Two matrices and are equal, written , exactly when:

7.2.3

Algebraic Operations on Matrices

Three algebraic operations are defined on matrices.

7.2.4

Properties of Matrix Addition, Scalar Multiplication and Product of Matrices

Let be matrices of orders that make the stated operation possible, and let be scalars.

7.2.5

Operation of Transpose of a Matrix and its Properties

The transpose of , written , is the matrix obtained by turning every row of into the corresponding column (equivalently, every column into the corresponding row): where , so the th entry of is the th…

7.2.6

Symmetric and Skew-Symmetric Matrices

A square matrix is:

+Exercise 7.1i24 questions
  1. Q1Construct an $m \times n$ matrix $A = [a_{ij}]$, where $a_{ij}$ is given by (i) $a_{ij} = \dfrac{(i-2j)^2}{2}$ with $m=2,\ n=3$ (ii) $a_{ij}…Free
  2. Q2Find the values of $p, q, r,$ and $s$ if $$\begin{pmatrix} p^2-1 & 0 & -31-q^3 \\ 7 & r+1 & 9 \\ -2 & 8 & s-1\end{pmatrix} = \begin{pmatrix}…Free
  3. Q3Determine the value of $x+y$ if $$\begin{pmatrix} 2x+y & 4x \\ 5x-7 & 4x\end{pmatrix} = \begin{pmatrix} 7 & 7y-13 \\ y & x+6\end{pmatrix}$$Free
  4. Q4Determine the matrices $A$ and $B$ if they satisfy $$2A - B + \begin{pmatrix} 6 & -6 & 0 \\ -4 & 2 & 1\end{pmatrix} = O \qquad \text{and} \q…Preview
  5. Q5If $A=\begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix}$, then compute $A^4$.Preview
  6. Q6Consider the matrix $A_\alpha = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{pmatrix}$. (i) Show that $A_\alpha A…Preview
  7. Q7If $A=\begin{pmatrix} 4 & 2 \\ -1 & x\end{pmatrix}$ and such that $(A-2I)(A-3I)=O$, find the value of $x$.Preview
  8. Q8If $A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{pmatrix}$, show that $A^2$ is a unit matrix.Preview
  9. Q9If $A=\begin{pmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{pmatrix}$ and $A^3-6A^2+7A+kI=O$, find the value of $k$.Preview
  10. Q10Give your own examples of matrices satisfying the following conditions in each case: (i) $A$ and $B$ such that $AB \ne BA$. (ii) $A$ and $B$…Preview
  11. Q11Show that $f(x)f(y)=f(x+y)$, where $f(x)=\begin{pmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{pmatrix}$.Preview
  12. Q12If $A$ is a square matrix such that $A^2=A$, find the value of $7A-(I+A)^3$.Preview
  13. Q13Verify the property $A(B+C)=AB+AC$, when the matrices $A, B,$ and $C$ are given by $$A=\begin{pmatrix} 2 & 0 & -3 \\ 1 & 4 & 5\end{pmatrix},…Preview
  14. Q14Find the matrix $A$ which satisfies the matrix relation $$A\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\end{pmatrix} = \begin{pmatrix} -7 & -8 & -…Preview
  15. Q15If $A^T=\begin{pmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3\end{pmatrix}$ and $B=\begin{pmatrix} 2 & -1 & 1 \\ 7 & 5 & -2\end{pmatrix}$, verify the fol…Preview
  16. Q16If $A$ is a $3\times4$ matrix and $B$ is a matrix such that both $A^TB$ and $BA^T$ are defined, what is the order of the matrix $B$?Preview
  17. Q17Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix: (i) $\begin{pmatrix} 4 & -2 \\ 3 & -5\end{pmatr…Preview
  18. Q18Find the matrix $A$ such that $$egin{pmatrix} 2 & -1 \ 1 & 0 \ -3 & 4\end{pmatrix} A^T = egin{pmatrix} -1 & -8 & -10 \ 1 & 2 & -5 \ 9 & 22…Preview
  19. Q19If $A=\begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y\end{pmatrix}$ is a matrix such that $AA^T=9I$, find the values of $x$ and $y$.Preview
  20. Q20(i) For what value of $x$ is the matrix $A=\begin{pmatrix} 0 & 1 & -2 \\ -1 & 0 & x^3 \\ 2 & -3 & 0\end{pmatrix}$ skew-symmetric? (ii) If $\…Preview
  21. Q21Construct the matrix $A=[a_{ij}]_{3\times3}$, where $a_{ij}=i-j$. State whether $A$ is symmetric or skew-symmetric.Preview
  22. Q22Let $A$ and $B$ be two symmetric matrices. Prove that $AB=BA$ if and only if $AB$ is a symmetric matrix.Preview
  23. Q23If $A$ and $B$ are symmetric matrices of same order, prove that (i) $AB+BA$ is a symmetric matrix. (ii) $AB-BA$ is a skew-symmetric matrix.Preview
  24. Q24A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins and almonds. Pack-I contains 100 gm of cashew nuts, 100 gm o…Preview
7.3

Determinants

To every square matrix of order , we associate a single number called the determinant of , written (also or or ).

7.3.1

Determinants of Matrices of Different Order

Order 1. For , — simply the single entry.

7.3.2

Properties of Determinants

Property 1 (transpose). — since expanding by rows gives the same value as expanding by columns.

+Exercise 7.2i21 questions
  1. Q1Without expanding the determinant, prove that $$\begin{vmatrix} s & a^2 & b^2+c^2 \\ s & b^2 & c^2+a^2 \\ s & c^2 & a^2+b^2 \end{vmatrix} =…Free
  2. Q2Show that $$\begin{vmatrix} b+c & bc & b^2c^2 \\ c+a & ca & c^2a^2 \\ a+b & ab & a^2b^2 \end{vmatrix} = 0.$$Free
  3. Q3Prove that $$\begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \end{vmatrix} = 4a^2b^2c^2.$$Free
  4. Q4Prove that $$\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = abc\left(1+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\rig…Preview
  5. Q5Prove that $$\begin{vmatrix} \sec^2\theta & \tan^2\theta & 1 \\ \tan^2\theta & \sec^2\theta & -1 \\ 38 & 36 & 2 \end{vmatrix} = 0.$$Preview
  6. Q6Show that $$\begin{vmatrix} x+2a & y+2b & z+2c \\ x & y & z \\ a & b & c \end{vmatrix} = 0.$$Preview
  7. Q7Write the general form of a $3 \times 3$ skew-symmetric matrix and prove that its determinant is $0$.Preview
  8. Q8If $$\begin{vmatrix} a & b & a\alpha+b \\ b & c & b\alpha+c \\ a\alpha+b & b\alpha+c & 0 \end{vmatrix} = 0,$$ prove that $a, b, c$ are in G.…Preview
  9. Q9Prove that $$\begin{vmatrix} 1 & a & a^2-bc \\ 1 & b & b^2-ca \\ 1 & c & c^2-ab \end{vmatrix} = 0.$$Preview
  10. Q10If $a, b, c$ are $p^{\text{th}}$, $q^{\text{th}}$ and $r^{\text{th}}$ terms of an A.P, find the value of $$\begin{vmatrix} a & b & c \\ p &…Preview
  11. Q11Show that $$\begin{vmatrix} a^2+x^2 & ab & ac \\ ab & b^2+x^2 & bc \\ ac & bc & c^2+x^2 \end{vmatrix}$$ is divisible by $x^4$.Preview
  12. Q12If $a, b, c$ are all positive, and are $p^{\text{th}}$, $q^{\text{th}}$ and $r^{\text{th}}$ terms of a G.P., show that $$\begin{vmatrix} \lo…Preview
  13. Q13Find the value of $$\begin{vmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \end{vmatrix}$$ if $x, y,…Preview
  14. Q14If $A = \begin{bmatrix} \dfrac{1}{2} & \alpha \\ 0 & \dfrac{1}{2} \end{bmatrix}$, prove that $\displaystyle\sum_{k=1}^{n} \det(A^k) = \dfrac…Preview
  15. Q15Without expanding, evaluate the following determinants: (i) $\begin{vmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{vmatrix}$ (ii) $\b…Preview
  16. Q16If $A$ is a square matrix and $|A| = 2$, find the value of $|AA^T|$.Preview
  17. Q17If $A$ and $B$ are square matrices of order $3$ such that $|A| = -1$ and $|B| = 3$, find the value of $|3AB|$.Preview
  18. Q18If $\lambda = -2$, determine the value of $$\begin{vmatrix} 0 & 2\lambda & 1 \\ \lambda^2 & 0 & 3\lambda^2+1 \\ -1 & 6\lambda-1 & 0 \end{vma…Preview
  19. Q19Determine the roots of the equation $$\begin{vmatrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5x^2 \end{vmatrix} = 0.$$Preview
  20. Q20Verify that $\det(AB) = (\det A)(\det B)$ for $A = \begin{bmatrix} 4 & 3 & -2 \\ 1 & 0 & 7 \\ 2 & 3 & -5 \end{bmatrix}$ and $B = \begin{bmat…Preview
  21. Q21Using cofactors of elements of second row, evaluate $|A|$, where $A = \begin{bmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{bmatrix}$.Preview
7.3.3

Application of Factor Theorem to Determinants

Theorem 7.3 (Factor Theorem for determinants). If every entry of a square matrix is a polynomial in , and vanishes (equals ) at , then is a factor of .

+Exercise 7.3i6 questions
  1. Q1Show that $\begin{vmatrix} x & a & a \\ a & x & a \\ a & a & x \end{vmatrix} = (x-a)^2(x+2a)$.Free
  2. Q2Show that $\begin{vmatrix} b+c & a-c & a-b \\ b-c & c+a & b-a \\ c-b & c-a & a+b \end{vmatrix} = 8abc$.Free
  3. Q3Solve $\begin{vmatrix} x+a & b & c \\ a & x+b & c \\ a & b & x+c \end{vmatrix} = 0$.Preview
  4. Q4Show that $\begin{vmatrix} b+c & a & a^2 \\ c+a & b & b^2 \\ a+b & c & c^2 \end{vmatrix} = (a+b+c)(a-b)(b-c)(c-a)$.Preview
  5. Q5Solve $\begin{vmatrix} 4-x & 4+x & 4+x \\ 4+x & 4-x & 4+x \\ 4+x & 4+x & 4-x \end{vmatrix} = 0$.Preview
  6. Q6Show that $\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{vmatrix} = (x-y)(y-z)(z-x)$.Preview
7.3.4

Product of Determinants

Two determinants of the same order can be multiplied to yield a third determinant of that order, using any one of four equivalent schemes:

7.3.5

Relation between a Determinant and its Cofactor Determinant

Let , and let be the cofactors of respectively.

7.3.6

Area of a Triangle

For a triangle with vertices , the familiar shoelace area formula can be written compactly as the absolute value of a determinant: The modulus (absolute value) is essential: the raw determinant can co…

7.3.7

Singular and Non-Singular Matrices

31 Q

A square matrix is:

+Exercise 7.4i6 questions
  1. Q1Find the area of the triangle whose vertices are $(0, 0)$, $(1, 2)$ and $(4, 3)$.Free
  2. Q2If $(k, 2)$, $(2, 4)$ and $(3, 2)$ are vertices of the triangle of area $4$ square units then determine the value of $k$.Free
  3. Q3Identify the singular and non-singular matrices: (i) $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ (ii) $\begin{bmatri…Preview
  4. Q4Determine the values of $a$ and $b$ so that the following matrices are singular: (i) $A = \begin{bmatrix} 7 & 3 \\ -2 & a \end{bmatrix}$ (ii…Preview
  5. Q5If $\cos 2\theta = 0$, determine $\begin{vmatrix} 0 & \cos\theta & \sin\theta \\ \cos\theta & \sin\theta & 0 \\ \sin\theta & 0 & \cos\theta…Preview
  6. Q6Find the value of the product: $\begin{vmatrix} \log_3 64 & \log_4 3 \\ \log_3 8 & \log_4 9 \end{vmatrix} \times \begin{vmatrix} \log_2 3 &…Preview
+Exercise 7.5i25 questions
  1. Q1If $a_{ij} = \dfrac{1}{2}(3i - 2j)$ and $A = [a_{ij}]_{2\times 2}$ is (1) $\begin{bmatrix} \dfrac{1}{2} & 2 \\ -\dfrac{1}{2} & 1 \end{bmatri…Free
  2. Q2What must be the matrix $X$, if $2X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}$? (1) $\be…Free
  3. Q3Which one of the following is not true about the matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 5 \end{bmatrix}$? (1) a scalar ma…Free
  4. Q4If $A$ and $B$ are two matrices such that $A + B$ and $AB$ are both defined, then (1) $A$ and $B$ are two matrices not necessarily of same o…Preview
  5. Q5If $A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}$, then for what value of $\lambda$, $A^2 = O$? (1) $0$ (2) $\pm 1$ (3) $-…Preview
  6. Q6If $A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}$, $B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}$ and $(A + B)^2 = A^2 + B^2$, th…Preview
  7. Q7If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where $I$ is $3…Preview
  8. Q8If $A$ is a square matrix, then which of the following is not symmetric? (1) $A + A^T$ (2) $AA^T$ (3) $A^T A$ (4) $A - A^T$Preview
  9. Q9If $A$ and $B$ are symmetric matrices of order $n$, where $(A \neq B)$, then (1) $A + B$ is skew-symmetric (2) $A + B$ is symmetric (3) $A +…Preview
  10. Q10If $A = \begin{bmatrix} a & x \\ y & a \end{bmatrix}$ and if $xy = 1$, then $\det(A A^T)$ is equal to (1) $(a - 1)^2$ (2) $(a^2 + 1)^2$ (3)…Preview
  11. Q11The value of $x$, for which the matrix $A = \begin{bmatrix} e^{x-2} & e^{7+x} \\ e^{2+x} & e^{2x+3} \end{bmatrix}$ is singular is (1) $9$ (2…Preview
  12. Q12If the points $(x, -2),\ (5, 2),\ (8, 8)$ are collinear, then $x$ is equal to (1) $-3$ (2) $\dfrac{1}{3}$ (3) $1$ (4) $3$Preview
  13. Q13If $\begin{vmatrix} 2a & x_1 & y_1 \\ 2b & x_2 & y_2 \\ 2c & x_3 & y_3 \end{vmatrix} = \dfrac{abc}{2} \neq 0$, then the area of the triangle…Preview
  14. Q14If the square of the matrix $\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ is the unit matrix of order $2$, then $\alpha…Preview
  15. Q15If $\Delta = \begin{vmatrix} a & b & c \\ x & y & z \\ p & q & r \end{vmatrix}$, then $\begin{vmatrix} ka & kb & kc \\ kx & ky & kz \\ kp &…Preview
  16. Q16A root of the equation $\begin{vmatrix} 3-x & -6 & 3 \\ -6 & 3-x & 3 \\ 3 & 3 & -6-x \end{vmatrix} = 0$ is (1) $6$ (2) $3$ (3) $0$ (4) $-6$Preview
  17. Q17The value of the determinant of $A = \begin{bmatrix} 0 & a & -b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}$ is (1) $-2abc$ (2) $abc$ (3) $0$…Preview
  18. Q18If $x_1, x_2, x_3$ as well as $y_1, y_2, y_3$ are in geometric progression with the same common ratio, then the points $(x_1, y_1),\ (x_2, y…Preview
  19. Q19If $\lfloor \cdot \rfloor$ denotes the greatest integer less than or equal to the real number under consideration and $-1 \le x < 0,\ 0 \le…Preview
  20. Q20If $a \neq b, b, c$ satisfy $\begin{vmatrix} a & 2b & 2c \\ 3 & b & c \\ 4 & a & b \end{vmatrix} = 0$, then $abc =$ (1) $a + b + c$ (2) $0$…Preview
  21. Q21If $A = \begin{vmatrix} -1 & 2 & 4 \\ 3 & 1 & 0 \\ -2 & 4 & 2 \end{vmatrix}$ and $B = \begin{vmatrix} -2 & 4 & 2 \\ 6 & 2 & 0 \\ -2 & 4 & 8…Preview
  22. Q22If $A$ is skew-symmetric of order $n$ and $C$ is a column matrix of order $n \times 1$, then $C^T A C$ is (1) an identity matrix of order $n…Preview
  23. Q23The matrix $A$ satisfying the equation $\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} A = \begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}$ is (…Preview
  24. Q24If $A + I = \begin{bmatrix} 3 & -2 \\ 4 & 1 \end{bmatrix}$, then $(A + I)(A - I)$ is equal to (1) $\begin{bmatrix} -5 & -4 \\ 8 & -9 \end{bm…Preview
  25. Q25Let $A$ and $B$ be two symmetric matrices of same order. Then which one of the following statement is not true? (1) $A + B$ is a symmetric m…Preview

Sample & Board Papers

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

+Show 28 questions28 questions
  1. Q1The factor of the determinant $\begin{vmatrix} x+a & b & c \\ a & x+b & c \\ a & b & x+c \end{vmatrix}$ is: (a) $x + c$ (b) $x$ (c) $x - a +…Preview
  2. Q2Write the additive inverse of the matrix $\begin{bmatrix} 1 & 0 & 2 \\ -1 & 5 & 3 \\ 2 & -1 & 1 \end{bmatrix}$.Preview
  3. Q3Prove that $\begin{vmatrix} 2x+y & x & y \\ 2y+z & y & z \\ 2z+x & z & x \end{vmatrix} = 0$.Preview
  4. Q4(a) Prove that $\begin{vmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{vmatrix} = 2abc(a+b+c)^3$. **OR** (b)…Preview
  5. Q5The value of $x$, for which the matrix $A=\begin{bmatrix}e^{x-2} & e^{7+x}\\ e^{2+x} & e^{2x+3}\end{bmatrix}$ is singular, is: (a) 7 (b) 6 (…Preview
  6. Q6Which one of the following is not true about the matrix $\begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 5\end{bmatrix}$? (a) an upper triang…Preview
  7. Q7Define diagonal and scalar matrices.Preview
  8. Q8Prove that square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.Preview
  9. Q9(a) Prove that $\begin{vmatrix}1 & x^2 & x^3\\ 1 & y^2 & y^3\\ 1 & z^2 & z^3\end{vmatrix}=(x-y)(y-z)(z-x)(xy+yz+zx)$. **OR** (b) Evaluate: $…Preview
  10. Q10If $A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}$, $B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}$ and $(A+B)^2 = A^2 + B^2$, then…Preview
  11. Q11The value of the determinant of $A = \begin{bmatrix} 0 & a & -b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}$ is: (a) $-2abc$ (b) $abc$ (c) $0$…Preview
  12. Q12If $A = \begin{bmatrix} a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2 \end{bmatrix}$ and $a^2+b^2+c^2=1$ then find the value of $A^2$.Preview
  13. Q13(a) Show that $\begin{vmatrix} b+c & a & a^2 \\ c+a & b & b^2 \\ a+b & c & c^2 \end{vmatrix} = (a+b+c)(a-b)(b-c)(c-a)$ using Factor theorem.…Preview
  14. Q14If A and B are symmetric matrices of order n, where (A$\ne$B), then: (a) A+B is a diagonal matrix (b) A+B is skew-symmetric (c) A+B is a zer…Preview
  15. Q15The value of $x$, for which the matrix $A = \begin{bmatrix} e^{x-2} & e^{7+x} \\ e^{2+x} & e^{2x+3} \end{bmatrix}$ is singular: (a) 7 (b) 9…Preview
  16. Q16Find $|A|$ if $A = \begin{bmatrix} 0 & \sin\alpha & \cos\alpha \\ \sin\alpha & 0 & \sin\beta \\ \cos\alpha & -\sin\beta & 0 \end{bmatrix}$.Preview
  17. Q17Prove that $\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{vmatrix} = (x-y)(y-z)(z-x)$.Preview
  18. Q18(a) Show that $\begin{vmatrix} 2bc-a^2 & c^2 & b^2 \\ c^2 & 2ca-b^2 & a^2 \\ b^2 & a^2 & 2ab-c^2 \end{vmatrix} = \begin{vmatrix} a & b & c \…Preview
  19. Q19Which of the following is not true about the matrix $\begin{bmatrix}1 & 0 & 0\\0 & 0 & 0\\0 & 0 & 5\end{bmatrix}$? (a) an upper triangular m…Preview
  20. Q20If $A = \begin{bmatrix}1 & -2 & 3\\1 & 2 & 1\\x & 2 & -3\end{bmatrix}$ is singular, find the value of $x$.Preview
  21. Q21A root of the equation $\begin{vmatrix}3-x & -6 & 3\\ -6 & 3-x & 3\\ 3 & 3 & -6-x\end{vmatrix}=0$ is: (a) $0$ (b) $6$ (c) $-6$ (d) $3$Preview
  22. Q22Compute $|A|$ if $A=\begin{bmatrix}3 & 4 & 1\\ 0 & -1 & 2\\ 5 & -2 & 6\end{bmatrix}$Preview
  23. Q23(a) Using Factor theorem, prove that $\begin{vmatrix}b+c & a & a^2\\ c+a & b & b^2\\ a+b & c & c^2\end{vmatrix}=(a+b+c)(a-b)(b-c)(c-a)$ **OR…Preview
  24. Q24Construct an $m \times n$ matrix $A = [a_{ij}]$, where $a_{ij}$ is given by $a_{ij} = \dfrac{(i-2j)^2}{2}$ with $m=2, n=3$.Preview
  25. Q25Without expanding, evaluate $\begin{vmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{vmatrix}$Preview
  26. Q26If $A = \begin{bmatrix}\lambda & 1\\ -1 & -\lambda\end{bmatrix}$, then for what value of $\lambda$, $A^2=0$? (a) $-1$ (b) $0$ (c) $1$ (d) $\…Preview
  27. Q27If $A = \begin{bmatrix}0 & c & b\\ c & 0 & a\\ b & a & 0\end{bmatrix}$, compute $A^2$Preview
  28. Q28Evaluate $\begin{vmatrix}2026 & 2023 & 0\\ 2025 & 2022 & 1\\ 2024 & 2021 & 0\end{vmatrix}$Preview