Let $A = \left[ \begin{array} { c c c } 1 & 2 & 3 \\ 10 & 20 & 31 \\ 11 & 22 & k \end{array} \right]$ and $\mathbf { v } = \left[ \begin{array} { l } x \\ y \\ z \end{array} \right]$, where $k$ is a constant and $x , y , z$ are variables.
Statements
(37) Regardless of the value of $k$, the matrix $A$ is not invertible, i.e., there is no $3 \times 3$ matrix $B$ such that $B A =$ the $3 \times 3$ identity matrix. (38) There is a unique $k$ such that determinant of $A$ is 0. (39) The set of solutions $( x , y , z )$ of the matrix equation $A \mathbf { v } = \left[ \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right]$ is either a line or a plane containing the origin. (40) If the equation $A \mathbf { v } = \left[ \begin{array} { c } p \\ q \\ r \end{array} \right]$ has a solution, then it must be true that $q = 10 p$.

We consider the matrix $$A = A(\mu) = \begin{pmatrix} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{pmatrix}$$
Show that $A(\mu)$ is invertible for every real $\mu$.

We consider the matrix $$A = A(\mu) = \begin{pmatrix} 2-\mu & -1 & \mu \\ -1 & 2-\mu & \mu-1 \\ 0 & -1 & 1 \end{pmatrix}$$
Calculate $A(\mu)_{s}$ and show that $A(\mu)_{s}$ is singular for $\mu = 1, 1-\sqrt{3}, 1+\sqrt{3}$.

Show that, if $A$ and $B$ belong to $S_n^{++}(\mathbf{R})$, then: $$\forall t \in [0,1], \quad \operatorname{det}((1-t)A + tB) \geq \operatorname{det}(A)^{1-t} \operatorname{det}(B)^t$$ Justify that this inequality remains valid for $A$ and $B$ only in $S_n^+(\mathbf{R})$.

What can be deduced about the function $\ln \circ \det$ on $S_n^{++}(\mathrm{R})$?

Let $A \in S_n^{++}(\mathbf{R})$ and let $g : t \in \mathbf{R} \mapsto \operatorname{det}(I_n + tA)$. Express, for all $t \in \mathbf{R}$, $g(t)$ using the eigenvalues of $A$. Deduce that $g$ is of class $C^\infty$ on $\mathbf{R}$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that $\varphi_\alpha$ is twice differentiable at 0 and that $$\varphi_\alpha''(0) = \operatorname{det}^{-\alpha}(A)\left(\alpha \operatorname{Tr}^2(A^{-1}M) + \operatorname{Tr}\left((A^{-1}M)^2\right)\right).$$

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that, if $\varphi_\alpha''(0) > 0$, then there exists $\eta > 0$ such that for all $t \in ]-\eta, \eta[$, $$\frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM) \geq \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A) - \operatorname{Tr}(A^{-1}M) \operatorname{det}^{-\alpha}(A) t.$$

Show that, if $A$ and $B$ belong to $S _ { n } ^ { + + } ( \mathrm { R } )$, then:
$$\forall t \in [ 0,1 ] , \quad \operatorname { det } ( ( 1 - t ) A + t B ) \geq \operatorname { det } ( A ) ^ { 1 - t } \operatorname { det } ( B ) ^ { t }$$
Justify that this inequality remains valid for $A$ and $B$ only in $S _ { n } ^ { + } ( \mathbf { R } )$.

What can we deduce about the function $\ln \circ \det$ on $S _ { n } ^ { + + } ( \mathbf { R } )$ ?

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and let $g : t \in \mathbf { R } \mapsto \operatorname { det } \left( I _ { n } + t A \right)$. Express, for all $t \in \mathbf { R } , g ( t )$ using the eigenvalues of $A$. Deduce that $g$ is of class $C ^ { \infty }$ on $\mathbf { R }$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$ and $\varphi _ { \alpha } ( t ) = \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M )$. Show that $\varphi _ { \alpha }$ is twice differentiable at 0 and that
$$\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) = \operatorname { det } ^ { - \alpha } ( A ) \left( \alpha \operatorname { Tr } ^ { 2 } \left( A ^ { - 1 } M \right) + \operatorname { Tr } \left( \left( A ^ { - 1 } M \right) ^ { 2 } \right) \right) .$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. Show that, if $\varphi _ { \alpha } ^ { \prime \prime } ( 0 ) > 0$, then there exists $\eta > 0$, such that for all $t \in ] - \eta , \eta [$,
$$\frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M ) \geq \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A ) - \operatorname { Tr } \left( A ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A ) t$$

Let k be a positive real number and let
$$A = \left[ \begin{array} { c c c } 2 k - 1 & 2 \sqrt { k } & 2 \sqrt { k } \\ 2 \sqrt { k } & 1 & - 2 k \\ - 2 \sqrt { k } & 2 k & - 1 \end{array} \right] \text { and } B = \left[ \begin{array} { c c c } 0 & 2 k - 1 & \sqrt { k } \\ 1 - 2 k & 0 & 2 \sqrt { k } \\ - \sqrt { k } & - 2 \sqrt { k } & 0 \end{array} \right]$$
If $\operatorname { det } ( \operatorname { adj } \mathrm { A } ) + \operatorname { det } ( \operatorname { adj } \mathrm { B } ) = 10 ^ { 6 }$, then $[ \mathrm { k } ]$ is equal to [Note : adj M denotes the adjoint of a square matrix M and $[ \mathrm { k } ]$ denotes the largest integer less than or equal to k].

The total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{array}\right| = 10$ is

Let $$M = \left[ \begin{array} { l l l } 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{array} \right] \quad \text { and } \quad \operatorname { adj } M = \left[ \begin{array} { r r r } - 1 & 1 & - 1 \\ 8 & - 6 & 2 \\ - 5 & 3 & - 1 \end{array} \right]$$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?
(A) $a + b = 3$
(B) $\quad ( \operatorname { adj } M ) ^ { - 1 } + \operatorname { adj } M ^ { - 1 } = - M$
(C) $\operatorname { det } \left( \operatorname { adj } M ^ { 2 } \right) = 81$
(D) If $M \left[ \begin{array} { l } \alpha \\ \beta \\ \gamma \end{array} \right] = \left[ \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right]$, then $\alpha - \beta + \gamma = 3$

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is $-18$, then the value of the determinant of $A$ is $\_\_\_\_$

Let $M = \begin{pmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{pmatrix}$ and $\text{adj}(M) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
where $a$ and $b$ are real numbers. Which of the following statements is(are) TRUE?
(A) $(a+b)^2 = 9$
(B) $\det(\text{adj}(M^2)) = 81$
(C) $\text{adj}(\text{adj}(M)) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
(D) $\det(\text{adj}(2M)) = 2^8$

Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations $$x + 2y + 3z = \alpha$$ $$4x + 5y + 6z = \beta$$ $$7x + 8y + 9z = \gamma - 1$$ is consistent. Let $|M|$ represent the determinant of the matrix $$M = \begin{pmatrix} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ \gamma & 1 & 0 \end{pmatrix}.$$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0, 1, 0)$ from the plane $P$.
The value of $|M|$ is ____.

Let $\beta$ be a real number. Consider the matrix
$$A = \left( \begin{array} { c c c } \beta & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{array} \right)$$
If $A ^ { 7 } - ( \beta - 1 ) A ^ { 6 } - \beta A ^ { 5 }$ is a singular matrix, then the value of $9 \beta$ is $\_\_\_\_$ .

Let $A = \left[ \begin{array} { c c c } 5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5 \end{array} \right]$. If $\left| A ^ { 2 } \right| = 25$, then $| \alpha |$ equals
(1) $5 ^ { 2 }$
(2) 1
(3) $1 / 5$
(4) 5

If $D = \left| \begin{array} { c c c } 1 & 1 & 1 \\ 1 & 1 + x & 1 \\ 1 & 1 & 1 + y \end{array} \right|$ for $x \neq 0 , y \neq 0$ then $D$ is
(1) divisible by neither $x$ nor $y$
(2) divisible by both $x$ and $y$
(3) divisible by $x$ but not $y$
(4) divisible by $y$ but not $x$

If $P = \left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$, then $\alpha$ is equal to
(1) 5
(2) 0
(3) 4
(4) 11

Let $d \in R$, and $A = \left[ \begin{array} { c c c } - 2 & 4 + d & ( \sin \theta ) - 2 \\ 1 & ( \sin \theta ) + 2 & d \\ 5 & ( 2 \sin \theta ) - d & ( - \sin \theta ) + 2 + 2 d \end{array} \right] , \theta \in [ 0,2 \pi ]$. If the minimum value of $\operatorname { det } ( A )$ is 8, then a value of $d$ is:
(1) $2 ( \sqrt { 2 } + 2 )$
(2) $2 ( \sqrt { 2 } + 1 )$
(3) $- 5$
(4) $- 7$

Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + x + 1 = 0$. Then for $y \neq 0$ in $R , \left| \begin{array} { c c c } y + 1 & \alpha & \beta \\ \alpha & y + \beta & 1 \\ \beta & 1 & y + \alpha \end{array} \right|$ is equal to
(1) $y ^ { 3 }$
(2) $y \left( y ^ { 2 } - 1 \right)$
(3) $y ^ { 3 } - 1$
(4) $y \left( y ^ { 2 } - 3 \right)$