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

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For a square matrix $A$ of order 3, if $\lvert A\rvert = -3$, then $\lvert 4\,adj A\rvert$ is equal to

Which of the following statements are TRUE?

A. If the product of two matrices is a zero matrix, it is not necessary that one of the matrix is a zero matrix.

B. For any three matrices A, B and C, $(AB)C = A(CB)$.

C. Diagonal elements of a skew symmetric matrix can be non-zero.

D. Multiplication of diagonal matrices of same order is always commutative.

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If $P = \dfrac{1}{14}\begin{bmatrix} 3 & 2 \\ -4 & 2 \end{bmatrix}$ and $P^{-1} = \begin{bmatrix} x & -2 \\ 4 & y \end{bmatrix}$, then the value of $x$ and $y$ respectively are

If A is a $2 \times 3$ matrix and B is a matrix such that $A^{T}B$ and $BA^{T}$ are both defined. Then order of B is

The value of $\Delta = \begin{vmatrix} 2\sin40^\circ & 2\sin50^\circ \\ -3\cos40^\circ & 3\cos50^\circ \end{vmatrix}$ is:

If $\begin{vmatrix} 2x & 2 \\ 8 & x \end{vmatrix} = \begin{vmatrix} -4 & -2 \\ 4 & -2 \end{vmatrix}$ then $x =$

If A and B are two non-singular matrices of same order, then which of the following statements are not correct ?

A. AB is non singular.

B. $(A+B)^{-1} = B^{-1}+A^{-1}$.

C. adj (AB) = adj B. adj A.

D. AB is not invertible.

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If the sum of a matrix A and its transpose is $\begin{bmatrix} -6 & 4 \\ 4 & 10 \end{bmatrix}$, then A is:

For a square matrix $A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 4 \\ 1 & 2 & 0 \end{bmatrix}$, which of the following statements is/are TRUE ?

A. det (A) = 9.

B. $A_{21}.A_{33} = -12$, where $A_{ij}$ is the cofactor of $a_{ij}$

C. det (A) = $a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{31}$.

D. $\lvert AA'\rvert \neq \lvert A\rvert^2$

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The system of equation $x-y+z=4, x-2y-2z=9, 2x+y+\alpha z=1$ has unique solution then value(/s) of $\alpha$ is (/are) (where $\mathbb{R}$ is set of real numbers)

For matrix $A = \begin{bmatrix} 2 & -3 \\ 3 & 4 \end{bmatrix}$ the value of $A^2 - 6A$ is equal to:(where I is an identity matrix of order 2)

For a matrix $A = \begin{bmatrix} a & -1 & b \\ -2 & 1 & -1 \\ 1 & 0 & 2 \end{bmatrix}$, $C_{32} = -1$ and $C_{33} = 3$, where $C_{ij}$ is the cofactor of element $a_{ij}$ in $A$. Then

If $A = \begin{bmatrix} 1 & a \\ b & 2 \end{bmatrix}$ and $B = \begin{bmatrix} a & 1 \\ 2 & -3b \end{bmatrix}$ such that $\lvert AB\rvert = \lvert 2B\rvert$ and $ab \neq -\dfrac{2}{3}$ then:

A. $\lvert A\rvert = 2$

B. $\lvert A\rvert = 4$

C. $\lvert B\rvert = 2$

D. $\lvert B\rvert = 4$

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The value of $\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}$ is equal to

If $f(x) = \begin{vmatrix} 0 & x+a & x+b \\ x-a & 0 & x+c \\ x-b & x-c & 0 \end{vmatrix}$ then the value of $f(0)$ is equal to:

The number of diagonal matrices of order 3 with elements either 1 or 2, is:

If $A^2 = 8A + kI, A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix}, I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ then value of $k$ is:

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If $A = \begin{bmatrix}0 & a & -3 \\ 2 & 0 & -1 \\ b & 1 & 0\end{bmatrix}$ is a skew-symmetric matrix, then

If $\begin{bmatrix}3 & 1\\2 & 1\end{bmatrix}P\begin{bmatrix}2 & 1\\3 & 2\end{bmatrix}=\begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}$ then matrix P is

If $A=\begin{bmatrix}0 & b-a & c-a\\a-b & 0 & c-b\\a-c & b-c & 0\end{bmatrix}$ then the value of $\lvert A \rvert$ is

Let A be the area of the triangle with vertices (3, 8), (5, -1) and $(k, 2)$, $k \in \mathbb{Z}$ (set of integers). If $2A = 75$, then k is equal to:

If $A=\begin{bmatrix}k & 2\\2 & k\end{bmatrix}$ such that $\lvert A^3 \rvert = 125$ then which of the following statements are TRUE?

(A) $\lvert k.adj(3A) \rvert = 405$

(B) $\lvert k.adj(3A) \rvert = 2025$

(C) $\lvert adj(3A) \rvert = 95$

(D) $\lvert adj(3A) \rvert = 45$

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If $A=\begin{bmatrix}5 & 0 & 0\\0 & 5 & 0\\0 & 0 & 5\end{bmatrix}$ such that $A^5=\lambda^2 A$, then the value of $\lambda$ is

If $A=\begin{bmatrix}x+4 & 3\\2 & 1\end{bmatrix}, B=\begin{bmatrix}5 & 2x-1\\2 & x\end{bmatrix}$ and $\lvert adj(AB) \rvert=32$, then the value of x is:

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If $A=\begin{bmatrix}1 & 2 & 2\\2 & 1 & -2\\\alpha & 2 & \beta\end{bmatrix}$ and $AA^T=9I$, where I is the identity matrix of order 3, then $(\beta-\alpha)$ equals:

Which of the following statements are true?

(A) A matrix which is both symmetric as well as skew-symmetric is a null matrix.

(B) All positive integral powers of a symmetric matrix are symmetric.

(C) If A is a square matrix, then $A+A^T$ is a symmetric matrix.

(D) All positive even integral powers of a skew-symmetric matrix are skew-symmetric.

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Match List-I with List-II

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