$$\begin{aligned} & A = \left[ \begin{array} { l l } 2 & 3 \\ 1 & 2 \end{array} \right] \\ & B = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 5 \end{array} \right] \end{aligned}$$
With the matrix notation
$$( 2 A - B ) \cdot \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$$
Which of the following is the system of linear equations?
A) $\begin{aligned} & x - 4 y = 0 \\ & 2 x - y = 1 \end{aligned}$
B) $\begin{aligned} & x + 2 y = 0 \\ & 2 x - 3 y = 1 \end{aligned}$
C) $\begin{aligned} & 2 x + y = 1 \\ & x - y = 0 \end{aligned}$
D) $\begin{aligned} & 3 x - 2 y = 1 \\ & 2 x + y = 0 \end{aligned}$
E) $\begin{aligned} & 3 x + 4 y = 1 \\ & 2 x - y = 0 \end{aligned}$

$$A = \left[ \begin{array} { l l } 3 & 2 \\ 0 & 1 \end{array} \right]$$
Given this, what is the value of the determinant $\left| A - A ^ { \top } \right|$?
A) 3
B) 4
C) 5
D) 6
E) 9

Let m be a positive real number and $u = \left[ \begin{array} { l l } x & y \end{array} \right]$. Given that
$$\mathrm { u } \cdot \left[ \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right] = \mathrm { u } \cdot \left[ \begin{array} { c c } \mathrm { m } & 0 \\ 0 & \mathrm {~m} \end{array} \right]$$
the matrix equation has infinitely many solutions for u, what is m?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) 3
E) 4

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) $\left[ \begin{array} { l l l } 6 & 4 & 12 \end{array} \right]$

Let I be the $2 \times 2$ identity matrix and
$$A = \left[ \begin{array} { l l } 4 & 5 \\ 1 & 3 \end{array} \right]$$
Accordingly, which of the following is $( \mathbf { A } - \mathbf { I } ) ^ { - \mathbf { 1 } }$ equal to?
A) $\left[ \begin{array} { r r } 2 & - 5 \\ - 1 & 3 \end{array} \right]$
B) $\left[ \begin{array} { r r } 1 & - 4 \\ - 2 & 3 \end{array} \right]$
C) $\left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 4 \end{array} \right]$
D) $\left[ \begin{array} { l l } - 2 & 5 \\ - 1 & 0 \end{array} \right]$
E) $\left[ \begin{array} { l l } 2 & - 5 \\ 0 & - 3 \end{array} \right]$

$$\left[ \begin{array} { l l } 3 & 2 \\ 1 & 0 \end{array} \right] \cdot \mathrm { A } = \left[ \begin{array} { c c } - 2 & 4 \\ 1 & 5 \end{array} \right]$$
What is the determinant of matrix A that satisfies this equation?
A) 4
B) 5
C) 6
D) 7
E) 8

Let $A$ and $B$ be $2 \times 1$ matrices and $t$ be a variable such that for all $x$ and $y$ values satisfying
$$x - y = 3$$
we have
$$\left[ \begin{array} { l } x \\ y \end{array} \right] = t A + B$$
Accordingly, which of the following could matrices A and B be, respectively?
A) $\left[ \begin{array} { l } 1 \\ 0 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
B) $\left[ \begin{array} { l } 0 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
C) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 1 \end{array} \right]$
D) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 0 \end{array} \right]$
E) $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ - 3 \end{array} \right]$

The inverse of matrix A is $A ^ { - 1 } = \left[ \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right]$. Given that
$$A \cdot \left[ \begin{array} { l } 1 \\ a \end{array} \right] = \left[ \begin{array} { l } b \\ 4 \end{array} \right]$$
what is the sum $\mathrm { a } + \mathrm { b }$?
A) 5
B) 7
C) 8
D) 9
E) 11

$$A = \left[ \begin{array} { r r } 1 & 0 \\ - 1 & 3 \end{array} \right], \quad B = \left[ \begin{array} { r r } - 1 & 1 \\ 0 & m \end{array} \right]$$
The matrices satisfy the equality
$$\operatorname { det } ( A + B ) = \operatorname { det } ( A ) + \operatorname { det } ( B )$$
Accordingly, what is m?
A) - 3
B) - 1
C) 0
D) 2
E) 4

$$3 x - y = 2$$ $$5 x + 2 y = 3$$
The matrix representation of the linear equation system is
$$A \cdot \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { l } 2 \\ 3 \end{array} \right]$$
Given that
$$A \cdot \left[ \begin{array} { l } 1 \\ 2 \end{array} \right] = \left[ \begin{array} { l } a \\ b \end{array} \right]$$
what is the sum $\mathbf { a + b }$?
A) 4
B) 6
C) 8
D) 10
E) 12