We are given matrices $A$, $B$, and $C$, and asked to evaluate the expression:

$(3)BA + (4)AC$

where:

• $A$ is a $2 \times 3$ matrix,
• $B$ is a $2 \times 2$ matrix,
• $C$ is a $3 \times 3$ matrix.

Let’s start by analyzing each matrix multiplication:

Matrices:

1. $A = \begin{bmatrix} -2 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix}$
2. $B = \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}$
3. $C = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 2 & 1 \\ -2 & 1 & -3 \end{bmatrix}$

Step 1: Evaluate $BA$

• Matrix $B$ is $2 \times 2$, and matrix $A$ is $2 \times 3$. Matrix multiplication between $B$ and $A$ is not possible, as the number of columns in $B$ (2) does not match the number of rows in $A$ (2).

Thus, $BA$ cannot be computed, and we conclude that the expression is not valid. Hence, the problem cannot be solved as written.

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Would you like to explore something else or clarify the problem further? Here are some related questions to expand your understanding:

1. What are the conditions for matrix multiplication to be possible?
2. How do you handle matrix multiplication for non-square matrices?
3. Can you provide an example where the multiplication of matrices results in a non-square matrix?
4. What is the difference between left and right matrix multiplication?
5. How do you compute the product of a scalar and a matrix?

Tip: Always check the dimensions of the matrices to ensure multiplication is feasible before proceeding with operations.