Let A be a square matrix such that $\mathrm { AA } ^ { \mathrm { T } } = \mathrm { I }$. Then $\frac { 1 } { 2 } \mathrm { ~A} \left[ \left( \mathrm { ~A} + \mathrm { A } ^ { \mathrm { T } } \right) ^ { 2 } + \left( \mathrm { A } - \mathrm { A } ^ { \mathrm { T } } \right) ^ { 2 } \right]$ is equal to
(1) $A ^ { 2 } + I$
(2) $A ^ { 3 } + I$
(3) $A ^ { 2 } + A ^ { T }$
(4) $\mathrm { A } ^ { 3 } + \mathrm { A } ^ { \mathrm { T } }$

If $\mathrm { A } , \mathrm { B }$, and $\left( \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) \right)$ are non-singular matrices of same order, then the inverse of $\mathrm { A } \left( \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) \right) ^ { - 1 } \mathrm {~B}$, is equal to
(1) $\mathrm { AB } ^ { - 1 } + \mathrm { A } ^ { - 1 } \mathrm {~B}$
(2) $\operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right)$
(3) $\frac { A B ^ { - 1 } } { | A | } + \frac { B A ^ { - 1 } } { | B | }$
(4) $\frac { 1 } { | A B | } ( \operatorname { adj } ( B ) + \operatorname { adj } ( A ) )$

Consider a real $2 \times 2$ matrix $\left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$. If $\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right] \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right] \left[ \begin{array} { c c } 1 & 0 \\ 0 & - 2 \end{array} \right] = \left[ \begin{array} { c c } 3 & - 4 \\ - 9 & - 7 \end{array} \right]$, what is the value of $c - 2b$?
(1) $- 11$ (2) $- 4$ (3) $1$ (4) $10$ (5) $11$

$$A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$$
Given that $A^{t}$ is the transpose of the matrix and $A^{-1}$ is its inverse matrix, which of the following is the product $A^{t} \cdot A^{-1}$?
A) $\begin{bmatrix} \frac{5}{2} & -3 \\ \frac{9}{2} & -5 \end{bmatrix}$
B) $\begin{bmatrix} \frac{3}{2} & -2 \\ 1 & 3 \end{bmatrix}$
C) $\begin{bmatrix} -2 & \frac{-9}{2} \\ 3 & \frac{5}{2} \end{bmatrix}$
D) $\begin{bmatrix} \frac{9}{2} & 3 \\ \frac{-5}{2} & -1 \end{bmatrix}$
E) $\begin{bmatrix} -3 & -1 \\ \frac{5}{2} & -2 \end{bmatrix}$

$$A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right], \quad B = \left[ \begin{array} { l l } 1 & 0 \\ \cdots & \cdots \end{array} \right]$$

Let a, b and c be positive real numbers,
$$\left[ \begin{array} { l l } a & b \\ 0 & c \end{array} \right] \cdot \left[ \begin{array} { l l } a & b \\ 0 & c \end{array} \right] = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right]$$
The matrix equation is given. Accordingly, what is the sum $a + b + c$?
A) $\frac { 11 } { 3 }$
B) $\frac { 7 } { 4 }$
C) 4
D) 5
E) 6

Let A be a $3 \times 3$ matrix. Given that
$$\begin{aligned} & { \left[ \begin{array} { l l l } 2 & 1 & 3 \end{array} \right] \cdot A = \left[ \begin{array} { l l l } 0 & 2 & 2 \end{array} \right] } \\ & { \left[ \begin{array} { l l l } 1 & 4 & 0 \end{array} \right] \cdot A = \left[ \begin{array} { l l l } 3 & 1 & 5 \end{array} \right] } \end{aligned}$$
What is the product $\left[ \begin{array} { l l l } 5 & 6 & 6 \end{array} \right] \cdot A$ equal to?
A) $\left[ \begin{array} { l l l } 2 & 1 & 3 \end{array} \right]$
B) $\left[ \begin{array} { l l l } 3 & 3 & 7 \end{array} \right]$
C) $\left[ \begin{array} { l l l } 3 & 5 & 9 \end{array} \right]$
D) $\left[ \begin{array} { l l l } 6 & 2 & 10 \end{array} \right]$
E) $\left[ \begin{array} { l l l } 6 & 4 & 12 \end{array} \right]$