Math Snap

STEP 1

1. Matrix addition and subtraction require matrices of the same dimensions.
2. Scalar multiplication can be performed on any matrix.
3. The given matrices $A$ and $B$ have dimensions $2 \times 3$ and $2 \times 2$, respectively.

STEP 2

1. Determine if $A + B$ is possible.
2. Determine if $A - B$ is possible.
3. Calculate $3A$.
4. Determine if $3A - 4B$ is possible.

STEP 3

To add matrices $A$ and $B$, they must have the same dimensions. Matrix $A$ is $2 \times 3$ and matrix $B$ is $2 \times 2$. Since they do not have the same dimensions, $A + B$ is not possible.
Result for $A + B$: IMPOSSIBLE

STEP 4

To subtract matrices $A$ and $B$, they must have the same dimensions. Matrix $A$ is $2 \times 3$ and matrix $B$ is $2 \times 2$. Since they do not have the same dimensions, $A - B$ is not possible.
Result for $A - B$: IMPOSSIBLE

STEP 5

To calculate $3A$, multiply each element of matrix $A$ by 3:
\[
3A = 3 \times \begin{bmatrix} 7 & 0 & 3 \\ -4 & -2 & 0 \end{bmatrix}
= \begin{bmatrix} 3 \times 7 & 3 \times 0 & 3 \times 3 \\ 3 \times (-4) & 3 \times (-2) & 3 \times 0 \end{bmatrix}
= \begin{bmatrix} 21 & 0 & 9 \\ -12 & -6 & 0 \end{bmatrix}$$

To calculate $3A - 4B$, both matrices must have the same dimensions. Since $A$ is $2 \times 3$ and $B$ is $2 \times 2$, they do not have the same dimensions, and $3A - 4B$ is not possible.
Result for $3A - 4B$: IMPOSSIBLE
The results are:
(a) $A + B$: IMPOSSIBLE
(b) $A - B$: IMPOSSIBLE
(c) $3A$:
$$\begin{bmatrix} 21 & 0 & 9 \\ -12 & -6 & 0 \end{bmatrix}$$ (d) $3A - 4B$: IMPOSSIBLE