Matrices and Determinants

Let $A = \begin{bmatrix} a & 1 \\ 1 & b \end{bmatrix}$, where $a$ and $b$ are the roots of the equation $x^2 - 4x + 2 = 0$. If $A + A^{-1} = kI_2$, then the value of $k$ is ____

If $A = \begin{bmatrix} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{bmatrix}$, $B = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$ such that $XA = B^T$ and $A^{-1}Y = B$, then $XY = $

Let $A = \begin{bmatrix} 3 & 1 & 2 \\ 1 & 2 & 0 \\ 1 & 1 & 4 \end{bmatrix}$ and $pC_{11} + 4C_{21} - 5C_{32} = -2$, where $C_{ij}$ denotes the cofactor of an element $a_{ij}$ of matrix $A$, then the value of $p$ is :

If $A = \begin{bmatrix} 1 & -5 \\ -2 & 4 \end{bmatrix}$, then $A^{-1} =$

The inverse of matrix $\begin{bmatrix} 1+pq & p & 0 \\ q & 1+pq & p \\ 0 & q & 1 \end{bmatrix}$ is ...

If $A = [a_{ij}]_{3\times3}$, where $a_{ij} = \begin{cases} 1, & \text{if } i+j \text{ is even} \\ 0, & \text{if } i+j \text{ is odd} \end{cases}$, then $\text{adj}(A) = \ldots$ ..

The inverse of the matrix $A = \begin{bmatrix} 2 & -1 & 4 \\ 4 & -3 & 1 \\ 1 & 2 & 1 \end{bmatrix}$ is $B = \dfrac{1}{37}\begin{bmatrix} -5 & 9 & 11 \\ -3 & -2 & 14 \\ 11 & -5 & k \end{bmatrix}$, then the value of $k$ is...

If matrix $A = \begin{bmatrix} -1 & 2025 & 2026 \\ 0 & 2 & 2027 \\ 0 & 0 & -1 \end{bmatrix}$, then the sum of all elements in $\text{adj}(A^{-1})$ is equal to...

Let $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & -13 \\ 5 & -10 \end{bmatrix}$ be two matrices. If the variables $x$ and $y$ satisfy the matrix equation $((A^{-1})^2 + B)\begin{bmatrix} x \\ y \end{bmatr...

Let $A = \begin{bmatrix} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$. If $B = \text{adj}\,A$, then the matrix $B^{-1}$ is equal to...

Let $A = \begin{bmatrix} -3 & 2 \\ 1 & 4 \end{bmatrix}$ and if $A^2 - 2A + I = \begin{bmatrix} 18 & p \\ q & 11 \end{bmatrix}$, then $\ldots$

If matrix A and its inverse $A^{-1}$ are given by $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} \dfrac{1}{2} & -\dfrac{1}{2} & \dfrac{1}{2} \\ -4 & 3 & y \\ \dfrac{5}{2} & -\dfrac{3}{2...

If $A$ is a non-singular matrix and $A^2 - A + I = 0$, then $A^{-1} = \ldots$

If the matrix $A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$ is expressed as the sum of a symmetric matrix B and a skew symmetric matrix C then which of the following relations is correct?

If $A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$; $B = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ then, $(2A + 3B)^{-1} =$ _______

If $A = \begin{bmatrix} 1 & -\tan\dfrac{\theta}{2} \\ \tan\dfrac{\theta}{2} & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & \tan\dfrac{\theta}{2} \\ -\tan\dfrac{\theta}{2} & 1 \end{bmatrix}$ then $A^{-1}B$ is equal to

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A\,(\text{adj }A) = AA^T$, Then $5a + b =$

Inverse of $\begin{bmatrix} 3 & -2 \\ 1 & 4 \end{bmatrix}$ is

If $A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}$ and $A^2 + xA + yI_2 = O_2$, where $I_2$ and $O_2$ are the identity matrix and null matrix of order 2 respectively, then:

If $A = [a_{ij}]_{3 \times 3}$ is a matrix such that $a_{ij} = |2i - 5j|$, where $|.|$ denotes the modulus function, then the element in the $2^{\text{nd}}$ row and $3^{\text{rd}}$ column of $A^{-1}$ is ...

If $A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$, $B^{-1} = \begin{bmatrix} \dfrac{1}{5} & \dfrac{2}{5} \\ \dfrac{2}{5} & -\dfrac{1}{5} \end{bmatrix}$, then $(AB)^{-1} = $

If $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, then the matrix $A^{-3}$ when $\theta = \dfrac{\pi}{6}$ is equal to...

Let $A = \begin{bmatrix} 2k-1 & 1 & 1 \\ 0 & 2k-1 & 1 \\ 0 & 0 & 2k-1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2k-1 & 1 \\ 1-2k & 0 & k \\ -1 & -k & 0 \end{bmatrix}$ where k is a real number. If $\det(\text{adj} A) + \det(\text{adj} B) =...

The element in the third row and the second column in the inverse matrix of a matrix $\begin{bmatrix} 1 & 3 & 3 \\ 3 & 1 & 3 \\ 3 & 3 & 4 \end{bmatrix}$ is

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, $C = \begin{bmatrix} 7 & 3 \\ 0 & 6 \end{bmatrix}$ and $AB = C$, then the inverse of matrix B is

If $A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}_{3 \times 3}$, and $A^{-1} = \begin{bmatrix} \gamma & -1 & 1 \\ \alpha & 6 & -5 \\ \beta & -2 & 2 \end{bmatrix}_{3 \times 3}$, then $|\alpha \cdot \beta \cdot \gamma...

Let $A = \begin{bmatrix} -5 & -3 \\ 2 & 1 \end{bmatrix}$. The Row transformation $R_1 \rightarrow R_1 + 3R_2$ will transform matrix A into

The element in 1st row and 2nd column of the inverse of the matrix $\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix}$ is ...

If $A = \begin{bmatrix} \sec\theta & -\tan\theta \\ -\tan\theta & \sec\theta \end{bmatrix}$, $\theta \in \left(0, \dfrac{\pi}{2}\right)$ such that $A + adj\,A = 4I$, then $\theta =$

If $A = \begin{bmatrix} 2i & i^3 \\ i^2 & 1 \end{bmatrix}$, then $A^{-1}$ is equal to

If $A = \dfrac{1}{2}\begin{bmatrix} -1 & -\sqrt{3} \\ \sqrt{3} & -1 \end{bmatrix}$ then $A^{-1} - A^2$ is not

If $\begin{bmatrix} 2 & 1 & -1 \\ -1 & 2 & 1 \\ 1 & -1 & 2 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ -2 \end{bmatrix}$ then the distance of point $P(a, b, c)$ from the plane $2x + y + 2z = 10$ is

If $A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix}$ then which of the following is true ?

The vectors $\bar{p}=\hat{i}+a \hat{j}+a^2 \hat{k}, \bar{q}=\hat{i}+b \hat{j}+b^2 \hat{k}$ and $\overline{\mathrm{r}}=\hat{\mathrm{i}}+\mathrm{c} \hat{\mathrm{j}}+\mathrm{c}^2 \hat{\mathrm{k}}$ are non-coplanar and $\left|\begin{array}{lll}...

If $A=\left[\begin{array}{rrr}1 & -2 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]$ then $A(I+\operatorname{adj} A)=$

Matrix A is non-singular matrix and $(A-3 I)(A-5 I)=0$, then $\frac{15}{8} A^{-1}=\ldots \ldots$

If $A$ is a matrix of order 2 and $I$ is the identity matrix of order 2 such that $A^2-4 A+3 I=0$ then $(A+3 I)^{-1}=$

If $A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]_{3 \times 3}$, then $A^{-1}=$

If $\mathrm{A}=\left[\begin{array}{cc}5 \mathrm{a} & -\mathrm{b} \\ 3 & 2\end{array}\right]$ and A .adj $\mathrm{A}=\mathrm{AA}^{\mathrm{T}}$, then $5 \mathrm{a}+\mathrm{b}=$

If A and B are non-singular matrices of order 2 such that $\quad(A B)^{-1}=\frac{1}{6}\left[\begin{array}{cc}-1 & -3 \\ 2 & 3\end{array}\right] \quad$ and $A^{-1}=\frac{1}{3}\left[\begin{array}{cc}4 & 3 \\ -1 & 0\end{array}\right]$ then $B^...

If $A=\left[\begin{array}{rrr}1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0\end{array}\right], B=\operatorname{adj} A$ and $C=5 A$, then $\frac{|\operatorname{adjB}|}{|\mathrm{C}|}=$

If $A=\left[\begin{array}{cc}1 & \cot \frac{\theta}{2} \\ -\cot \frac{\theta}{2} & 1\end{array}\right]$ then $A^{-1}=$

If matrix $\quad A=\frac{1}{11}\left[\begin{array}{rrr}-1 & 7 & -24 \\ 2 & a & 4 \\ 2 & -3 & 15\end{array}\right] \quad$ and $A^{-1}=\left[\begin{array}{rrr}3 & 3 & 4 \\ 2 & -3 & 4 \\ b & -1 & c\end{array}\right]$, then the values of $a, b,...

The third element in the second row of adjoint of a matrix $A=\left[a_{i j}\right]_{3 \times 3}$ (where $a_{i j}=2 i+j$ ) is

Let A be a non-singular matrix of order n and $|A|=k$, then $(\operatorname{adj} A)^{-1}$ is

If $A=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right]$, then $A^T A^{-1}=$

If $\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 4 \\ 1 & 3 & 4\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}12 \\ 15 \\ 13\end{array}\right]$, then the value of $x^2+y^2+z^2=$

If $A=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]$,
where $A_{21}, A_{22}, A_{23}$ are cofactors of $a_{21}, a_{22}, a_{23}$ respectively, then the value of $\mathrm...

If $A=\left[\begin{array}{rr}1 & 2 \\ -1 & 4\end{array}\right]$ and $A^{-1}=\alpha I+\beta A \alpha, \beta \in R$ where I is the identity matrix of order 2 , then $4(\alpha+\beta)=$

If $A=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & a & 3 \\ 3 & 2 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2 & 0 & b \\ 7 & -1 & -2 \\ c & 1 & 1\end{array}\right]$ and if matrix $B$ is the inverse of matrix $A$, then value of $4 a...