Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show more generally that $$\operatorname{det}\left(A + \mathbf{u v}^T\right) = \operatorname{det}(A)\left(1 + \left\langle \mathbf{v}, A^{-1} \mathbf{u} \right\rangle\right).$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Let $C \in \mathcal{M}_n(\mathbb{R})$ be a matrix such that $\operatorname{det}(C) = 0$. Is it always true that $\operatorname{det}\left(C + \mathbf{u v}^T\right) = 0$? Justify your answer.

We consider $A \in \mathcal{S}_n(\mathbb{R})$ symmetric with eigenvalues $\lambda_1 \leqslant \cdots \leqslant \lambda_n$ and corresponding orthonormal basis of eigenvectors $\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$. We set $B = A + \mathbf{u u}^T$ with $\|\mathbf{u}\| = 1$. We denote by $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$ the characteristic polynomial of $A$, and $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$ that of $B$. Show that, for all $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$, we have $$\chi_B(x) = \chi_A(x)\left(1 - \sum_{k=1}^n \frac{\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle^2}{x - \lambda_k}\right).$$

Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.

In this question, we assume that $M$ is invertible. Prove that $m = n$ and that $A$ and $B$ are invertible.

A ``basic row operation'' on a matrix means adding a multiple of one row to another row. Consider the matrices $$A = \left(\begin{array}{rrr} x & 5 & x \\ 1 & 3 & -2 \\ -2 & -2 & 2 \end{array}\right) \quad \text{and} \quad B = \left(\begin{array}{rrr} 0 & 0 & 21 \\ 1 & -1 & -14 \\ 0 & \frac{4}{3} & 4 \end{array}\right)$$ It is given that $B$ can be obtained from $A$ by applying finitely many basic row operations. Then, the value of $x$ is:
(A) 3
(B) $-3$
(C) $-1$
(D) 2.

Let $n \geq 3$. Let $A = \left( \left( a _ { i j } \right) \right) _ { 1 \leq i , j \leq n }$ be an $n \times n$ matrix such that $a _ { i j } \in \{ 1 , - 1 \}$ for all $1 \leq i , j \leq n$. Suppose that $$\begin{aligned} & a _ { k 1 } = 1 \text { for all } 1 \leq k \leq n \text { and } \\ & \sum _ { k = 1 } ^ { n } a _ { k i } a _ { k j } = 0 \text { for all } i \neq j \end{aligned}$$ Show that $n$ is a multiple of 4.

Let $M$ be a $3 \times 3$ matrix with all entries being 0 or 1 . Then, all possible values for $\operatorname { det } ( M )$ are
(A) $0 , \pm 1$
(B) $0 , \pm 1 , \pm 2$
(C) $0 , \pm 1 , \pm 3$
(D) $0 , \pm 1 , \pm 2 , \pm 3$.

Consider all $2 \times 2$ matrices whose entries are distinct and taken from the set $\{ 1,2,3,4 \}$. The sum of determinants of all such matrices is
(A) 24 .
(B) 10 .
(C) 12 .
(D) 0 .

Define a polynomial $f ( x )$ by $$f ( x ) = \left| \begin{array} { l l l } 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{array} \right|$$ for all $x \in \mathbb { R }$, where the right hand side above is a determinant. Then the roots of $f ( x )$ are of the form
(A) $\alpha , \beta \pm i \gamma$ where $\alpha , \beta , \gamma \in \mathbb { R } , \gamma \neq 0$ and $i$ is a square root of $- 1$.
(B) $\alpha , \alpha , \beta$ where $\alpha , \beta \in \mathbb { R }$ are distinct.
(C) $\alpha , \beta , \gamma$ where $\alpha , \beta , \gamma \in \mathbb { R }$ are all distinct.
(D) $\alpha , \alpha , \alpha$ for some $\alpha \in \mathbb { R }$.

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname { det } ( \mathrm { A } )$ divisible by p is
A) $( p - 1 ) ^ { 2 }$
B) $2 ( p - 1 )$
C) $( p - 1 ) ^ { 2 } + 1$
D) $2 p - 1$

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that the trace of $A$ is not divisible by $p$ but $\operatorname { det } ( A )$ is divisible by $p$ is [Note : The trace of a matrix is the sum of its diagonal entries.]
A) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - \mathrm { p } + 1 \right)$
B) $\mathrm { p } ^ { 3 } - ( \mathrm { p } - 1 ) ^ { 2 }$
C) $( p - 1 ) ^ { 2 }$
D) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - 2 \right)$

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that $\operatorname { det } ( A )$ is not divisible by $p$ is
A) $2 p ^ { 2 }$
B) $p ^ { 3 } - 5 p$
C) $p ^ { 3 } - 3 p$
D) $p ^ { 3 } - p ^ { 2 }$

Let $\omega$ be the complex number $\cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$. Then the number of distinct complex numbers $z$ satisfying $\left| \begin{array} { c c c } z + 1 & \omega & \omega ^ { 2 } \\ \omega & z + \omega ^ { 2 } & 1 \\ \omega ^ { 2 } & 1 & z + \omega \end{array} \right| = 0$ is equal to

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $$\left[ \begin{array} { c c c } 1 & a & b \\ \omega & 1 & c \\ \omega ^ { 2 } & \omega & 1 \end{array} \right]$$ where each of $a , b$, and $c$ is either $\omega$ or $\omega ^ { 2 }$. Then the number of distinct matrices in the set $S$ is
(A) 2
(B) 6
(C) 4
(D) 8

Which of the following values of $\alpha$ satisfy the equation $$\left| \begin{array} { c c c } ( 1 + \alpha ) ^ { 2 } & ( 1 + 2 \alpha ) ^ { 2 } & ( 1 + 3 \alpha ) ^ { 2 } \\ ( 2 + \alpha ) ^ { 2 } & ( 2 + 2 \alpha ) ^ { 2 } & ( 2 + 3 \alpha ) ^ { 2 } \\ ( 3 + \alpha ) ^ { 2 } & ( 3 + 2 \alpha ) ^ { 2 } & ( 3 + 3 \alpha ) ^ { 2 } \end{array} \right| = - 648 \alpha$$?
(A) $-4$
(B) $9$
(C) $-9$
(D) $4$

How many $3 \times 3$ matrices $M$ with entries from $\{ 0,1,2 \}$ are there, for which the sum of the diagonal entries of $M ^ { T } M$ is 5 ?
[A] 126
[B] 198
[C] 162
[D] 135

Let $R = \left\{ \left( \begin{array} { c c c } a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{array} \right) : a , b , c , d \in \{ 0,3,5,7,11,13,17,19 \} \right\}$. Then the number of invertible matrices in $R$ is

Let $S = \left\{ A = \left( \begin{array} { l l l } 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{array} \right) : a , b , c , d , e \in \{ 0,1 \} \right.$ and $\left. | A | \in \{ - 1,1 \} \right\}$, where $| A |$ denotes the determinant of $A$. Then the number of elements in $S$ is $\_\_\_\_$ .

Let $I = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $P = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)$. Let $Q = \left( \begin{array} { l l } x & y \\ z & 4 \end{array} \right)$ for some non-zero real numbers $x , y$, and $z$, for which there is a $2 \times 2$ matrix $R$ with all entries being non-zero real numbers, such that $Q R = R P$.
Then which of the following statements is (are) TRUE?

(A)	The determinant of $Q - 2 I$ is zero
(B)	The determinant of $Q - 6 I$ is 12
(C)	The determinant of $Q - 3 I$ is 15
(D)	$y z = 2$

If $A = \left( \begin{array} { c } \alpha - 1 \\ 0 \\ 0 \end{array} \right) , B = \left( \begin{array} { c } \alpha + 1 \\ 0 \\ 0 \end{array} \right)$ be two matrices, then $A B ^ { T }$ is a non-zero matrix for $| \alpha |$ not equal to
(1) 2
(2) 0
(3) 1
(4) 3

$$\left|\begin{array}{ccc} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & b+c & -2c \end{array}\right| = \alpha(a+b)(b+c)(c+a) \neq 0$$
then $\alpha$ is equal to
(1) $a+b+c$
(2) $abc$
(3) 4
(4) 1

Let $P$ and $Q$ be $3 \times 3$ matrices with $P \neq Q$. If $P^{3} = Q^{3}$ and $P^{2}Q = Q^{2}P$, then the determinant of $(P^{2}+Q^{2})$ is equal to
(1) $-2$
(2) 1
(3) 0
(4) $-1$

If $\alpha , \beta \neq 0 , f ( n ) = \alpha ^ { n } + \beta ^ { n }$ and $$\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1 + f(1) & 1 + f(2) & 1 + f(3) \\ 1 + f(2) & 1 + f(3) & 1 + f(4) \end{vmatrix} = K ( 1 - \alpha ) ^ { 2 } ( 1 - \beta ) ^ { 2 } ( \alpha - \beta ) ^ { 2 }$$, then $K$ is equal to
(1) 1
(2) - 1
(3) $\alpha \beta$
(4) $\frac { 1 } { \alpha \beta }$

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A$ adj $A = A A^T$, then $5a + b$ is equal to:
(1) $-1$
(2) $5$
(3) $4$
(4) $13$