If

$$A=\left[\begin{array}{ll}1 & 5 \\ 3 & 2\end{array}\right]$$

and

$$B=\left[\begin{array}{rr}2 & 3 \\ -1 & -2\end{array}\right]$$

, compute A B and B A, if possible.

Find E F and F E if

$$E=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$$

and

$$F=\left[\begin{array}{ll}e & f \\ g & h\end{array}\right]$$

.

Find A B and B A if

$$A=\left[\begin{array}{lll}a & b & c \\ d & e & f\end{array}\right]$$

and

$$B=\left[\begin{array}{ll}1 & 2 \\ 3 & 4 \\ 5 & 6\end{array}\right]$$

Use Cramer’s Rule to solve the system, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions.

$$\left\{\begin{array}{l}x+y+z=3 \\ 2 x+y+z=4 \\ 2 x+2 y+z=5\end{array}\right.$$

The system do not have unique solution. Solve the system, if possible.

$$\left\{\begin{array}{r}x-y+2 z=-1 \\ 2 x+2 y+3 z=-3 \\ x+3 y+z=-2\end{array}\right.$$

Perform the matrix operation $A^2=A \cdot A$, if possible, with

$$\begin{array}{lll}A=\left[\begin{array}{rrr}1 & 3 & -3 \\ 2 & 4 & 1 \\ -1 & 3 & 2\end{array}\right] & B=\left[\begin{array}{rrr}1 & 2 & 1 \\ 2 & -1 & 3\end{array}\right] \\ C=\left[\begin{array}{rr}2 & 3 \\ -1 & 2 \\ 3 & -2\end{array}\right] \quad & D=\left[\begin{array}{lll}-2 & 3 & 1 \\ -3 & 2 & 2\end{array}\right]\end{array}$$

If C is a $3 \times 4$ matrix and D is a $4 \times 3$ matrix, what are the dimensions of C D and D C ?

Determine whether the planes represented by these following equations represent the same plane, parallel planes, or neither:

$$3 x-y+2 z=9 \text { and } 9 x-3 y+6 z=27$$

Use Cramer’s Rule to solve the system, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions.

$$\left\{\begin{array}{l}x-2 y=3 \\ -2 x+4 y=-6\end{array}\right.$$

The system do not have unique solution. Solve the system, if possible.

$$\left\{\begin{array}{r}x-3 y-2 z=3 \\ y+4 z=5 \\ 0=4\end{array}\right.$$

Solve the system algebraically.

$$\left\{\begin{array}{l}x^2-2 y=0 \\ 2 y-x=0\end{array}\right.$$

If x is proportional to y, and y = 10x, what is the constant of proportionality?

Use the following matrices to solve:

$$\begin{array}{ll}A=\left[\begin{array}{rrr}1 & 3 & -2 \\ 3 & 1 & 4 \\ -5 & 3 & 6\end{array}\right] & B=\left[\begin{array}{llr}2 & 1 & -1 \\ 3 & 2 & 4\end{array}\right] \\ C=\left[\begin{array}{rr}1 & 3 \\ 2 & 1 \\ 3 & -1\end{array}\right] & D=\left[\begin{array}{rrr}2 & 3 & 1 \\ 3 & 4 & -1 \\ 2 & 5 & 1\end{array}\right] \\ E=\left[\begin{array}{rrr}9 & 2 & -7 \\ -5 & 0 & 5 \\ 7 & -4 & -1\end{array}\right] & F=\left[\begin{array}{lll}2 & 1 & 3 \\ 4 & 0 & 1\end{array}\right]\end{array}$$

Use letters to represent the matrix products that are defined.

Write the augmented matrix associated with the given system.

$$\left\{\begin{array}{rr}3 x-4 y+6 z= & 10 \\ 2 x-4 y-5 z= & -14 \\ x+2 y-3 z= & 0\end{array}\right.$$