CUET Mathematics: AlgebraMatrices & Determinants. Free, no login required.

If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $A^{-1}$ is equal to

If P, Q and R are matrices of order 2x3, 3x5 and 5x3 respectively. Then which of the following are valid?

(A) P Q R

(B) P R Q

(C) Q R

(D) R Q

(E) P R

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Let $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$ and $a_{ij}$ be any element of matrix A, i, j ∈ {1,2,3}, then which of the following are TRUE?

(A) Minor of $a_{23} = 16$

(B) Minor of $a_{23} = -8$

(C) Cofactor of $a_{23} = -16$

(D) Cofactor of $a_{23} = 8$

(E) Cofactor of $a_{13} = 4$

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Let $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} -4 & -2 \\ -8 & -4 \end{vmatrix}$. Then

(A) $x = -4$

(B) $x = -6$

(C) $x = 4$

(D) $x = 6$

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If A and B are square matrices of order 3 such that $|A| = 3$ and $|B| = -1$, then $|3AB|$ is equal to

Which of the following statements is (are) true?

(A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix

(B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix

(C) $B^T AB$ is a symmetric matrix if A is a skew-symmetric matrix

(D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix

(E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix

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For what value of k, the following system of equations has infinitely many solutions?

$x + 2y = 5, 3x + ky = 15$

If $A$ is a square matrix and $I$ is an identity matrix of same order such that $A^2 = A$, then $(2I + A)^2 - 5A$ is

For the system $\begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$ which of the following statements are correct?

(A) The system has no solution.

(B) The system is consistent.

(C) It has infinitely many solutions.

(D) It has a unique solution.

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If A is an invertible matrix of order 3 and the determinant of A is 9, then the determinant of $A^{-1}$ is:

If the matrix $\begin{bmatrix} 0 & 7 & -12 \\ -7 & 0 & -5 \\ 2a & 5 & 3b \end{bmatrix}$ is skew-symmetric, then the value of $(4a + 3b)$ is:

Match List-I with List-II

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Let A, B, C, D and E be matrices of order $2 \times n, 3 \times k, 2\times p, n \times 3$ and $p \times k$ respectively. Choose the correct statement(s) from the following?

(A) EB + DB will be defined if $k = 3, p = n$.

(B) EB + DB will be defined if $k = 2, p = 3$.

(C) If n = p = 2, then the order of the matrix $5A^2 - 3C$ is $2 \times 2$.

(D) If n = p, then the order of the matrix $5A^2 - 3C$ is $p \times k$.

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Let $A = [a_{ij}]$ be a square matrix of order 2 with elements either 0 or 1. Then the difference between the possible number of singular and non-singular matrices is

If A and B are invertible matrices of same order, then which one of the following is NOT true?

If the system of equations

$x + 2y + 3z = 10$

$-x + y + \lambda z = 20$

$2x + 3y + \lambda z = 0$

does not possess a unique solution, then $\lambda$ is equal to

If $\begin{bmatrix} 2a + b & a - 2b \\ 5c - d & 4c + 3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}$, then the value of $a + 2b - 3c + 4d$ is equal to

If A and B are square matrices of the same order, then (A+B) (A-B) is equal to

If A and B are matrices of same order, then $(AB^T - BA^T)$ is always

Match List-I with List-II

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If matrices $A = [1 \quad 2 \quad 3]$ and $B = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, then $BA$ is equal to:

Let $A = \begin{bmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{bmatrix}$, where $0 \leq \theta \leq 2\pi$. Then which of the following are true?

(A) $|A| = 2 + 2 \sin^{2} \theta$

(B) $|A| = 2 + \sin^{2} \theta$

(C) minimum value of $|A|$ is $1$

(D) maximum value of $|A|$ is $4$

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If the points $(2, -3)$, $(\lambda, -1)$ and $(0, 4)$ are collinear, then the value of $\lambda$ is

Let $A$ and $B$ are square matrices of order 3 such that $A + B = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$. If $A$ is a symmetric matrix, then the value of $|B|$ is

If $M = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ and $N = [7 \quad 1 \quad -4]$, then $(MN)^T$ will be equal to:

If $A = [a_{ij}]_{2×2}$ where $a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}$ and $I$ is the identity matrix of order 2, then $(A^2 - 3A + 4I)$ is

(A) Symmetric Matrix

(B) Skew-symmetric Matrix

(C) Non-singular Matrix

(D) Square Matrix

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The matrix $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ is a

(A) Null matrix

(B) Unit matrix

(C) Symmetric matrix

(D) Skew-symmetric matrix

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If $A = [a_{ij}]_{3×3}$ where $a_{ij} = \begin{cases} (-1)^{i+j} - 1, & i = j \\ (-1)^{i+j}, & i \neq j \end{cases}$ then the value of $A + A^T$ is:

Which of the following is NOT correct?

If $A$ and $B$ are symmetric matrices of the same order, then $A B-B A$ is a :