Consider the following general matrix equation:

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $a_1$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ $a_2$

$=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $m_{11}$ $m_{12}$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ $m_{21}$ $m_{22}$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $x_1$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ $x_2$

which can also be abbreviated as:
$$A = M X$$
By definition, the determinant of $M$ is given by $$\det\,(M) = m_{11}m_{22}-m_{12}m_{21}$$
The following questions are about the relationship between the determinant of M and the ability to solve the equation above for A in terms of X or for X in terms of A.
Check the boxes which make the statement correct:
If the $\det\,(M)\ne 0$ then

A. some values of $X$ will have more than one value of $A$ which satisfy the equation.
B. some values of $A$ (such as $A=0$ ) will allow more than one $X$ to satisfy the equation.
C. some values of $X$ will have no values of $A$ which satisfy the equation.
D. given any $X$ there is one and only one $A$ which will satisfy the equation.
E. given any $A$ there is one and only one $X$ which will satisfy the equation.
F. some values of $A$ will have no values of $X$ which will satisfy the equation.

Check the boxes which make the statement correct:
If the $\det\,(M) = 0$ then

A. some values of $A$ (such as $A=0$ ) will allow more than one $X$ to satisfy the equation.
B. given any $A$ there is one and only one $X$ which will satisfy the equation.
C. given any $X$ there is one and only one $A$ which will satisfy the equation.
D. some values of $A$ will have no values of $X$ which will satisfy the equation.
E. there is no value of $X$ which satisfies the equation when $A = 0$.

Check the conditions that guarantee that $\det\,(M) = 0$:

A. Given any $A$ the is one and only one $X$ which will satisfy the equation.
B. When $A=0$ there is more than one $X$ which satisfies the equation.
C. Given any $X$ there is one and only one $A$ which will satisfy the equation.
D. There is some value of $A$ for which no value of $X$ satisfies the equation.