Adjoint and inverse of a square matrix — practice questions

Question 1 · difficulty L2 · understanding

If the ad joint of a 3 $$\times$$ 3 matrix P is \begin{aligned}\left[ {\matrix{ 1 & 4 & 4 \cr 2 & 1 & 7 \cr 1 & 1 & 3 \cr } } \right]\end{aligned}, then the possible value(s) of the determinant of P is(are)

• A. $$-$$2
• B. $$-$$1
• C. 1
• D. 2

Question 2 · difficulty L3 · understanding

Let $A=\left[$$\begin{array}{ccc}-1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1\end{array}$$\right]$satisfy$\mathrm{A}^2+\alpha(\operatorname{adj}(\operatorname{adj}(\mathrm{A})))+\beta(\operatorname{adj}(\mathrm{A})(\operatorname{adj}(\operatorname{adj}(\mathrm{A}))))=\left[$$\begin{array}{ccc}2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1\end{array}$$\right]$for some$\alpha, \beta \in \mathbb{R}$. Then$(\alpha-\beta)^2$is equal to$\_\_\_\_$

Question 3 · difficulty L3 · understanding

Consider the matrices $A = $$\begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix}$$$and$B = $$\begin{bmatrix} 3 & 9 \\ 1 & 3 \end{bmatrix}$$$. If matrices$P$and$Q$are such that$PA = B$and$AQ = B$, then the absolute value of the sum of the diagonal elements of$2(P + Q)$ is ________.

Question 4 · difficulty L3 · understanding

Let $A=\left[$$\begin{array}{ccc}0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0\end{array}$$\right]$and$B$be a matrix such that$B(I-A)=I+A$. Then the sum of the diagonal elements of$\mathrm{B}^{\mathrm{T}} \mathrm{B}$is equal to$\_\_\_\_$

Question 5 · difficulty L3 · understanding

For some $\alpha, \beta \in \mathbf{R}$, let $A=\left[$$\begin{array}{ll}\alpha & 2 \\ 1 & 2\end{array}$$\right]$and$B=\left[$$\begin{array}{ll}1 & 1 \\ 1 & \beta\end{array}$$\right]$be such that$A^2-4 A+2 I=B^2-3 B+I=O$. Then$\left(\operatorname{det}\left(\operatorname{adj}\left(A^3-B^3\right)\right)\right)^2$is equal to$\_\_\_\_$ .

Question 6 · difficulty L3 · understanding

Let $S=\left\{m \in \mathbf{Z}: A^{m^2}+A^m=3 I-A^{-6}\right\}$, where $A=\left[$$\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}$$\right]$. Then$n(S)$ is equal to __________.

Question 7 · difficulty L3 · understanding

Let $A$ be a square matrix of order 3 such that $\operatorname{det}(A)=-2$ and $\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=2^{m+n} \cdot 3^{m n}, m>n$. Then $4 m+2 n$ is equal to __________.

Question 8 · difficulty L3 · understanding

Let $$A$$ be a $$2 \times 2$$ symmetric matrix such that $$A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$$ and the determinant of $$A$$ be 1 . If $$A^{-1}=\alpha A+\beta I$$, where $$I$$ is an identity matrix of order $$2 \times 2$$, then $$\alpha+\beta$$ equals _________.

Question 9 · difficulty L3 · understanding

The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $$a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}$$, such that $$A=A^{-1}$$, is ___________.

Question 10 · difficulty L3 · understanding

Let \begin{aligned}X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right],\,Y = \alpha I + \beta X + \gamma {X^2}\end{aligned} and $$Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma ){X^2}$$, $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R. If \begin{aligned}{Y^{ - 1}} = \left[ {\matrix{ {{1 \over 5}} & {{{ - 2} \over 5}} & {{1 \over 5}} \cr 0 & {{1 \over 5}} & {{{ - 2} \over 5}} \cr 0 & 0 & {{1 \over 5}} \cr } } \right]\end{aligned}, then ($$\alpha$$ $$-$$ $$\beta$$ + $$\gamma$$) 2 is equal to ____________.