Skill: Simplify expressions involving matrix addition, subtraction, and scalar multiplication

To simplify an expression with matrices, first check that the matrices have the same dimensions. Then distribute any scalars across every term inside parentheses, combine coefficients of identical matrices, and calculate corresponding entries.

Suppose

$$A=\begin{bmatrix}1&2\\-1&3\end{bmatrix}, \qquad B=\begin{bmatrix}4&-2\\0&5\end{bmatrix}.$$

Simplify $2A-3(B-A)+B$.

Distribute $-3$ through the parentheses. Keep the subtraction order in $B-A$:

$$2A-3(B-A)+B =2A-3B+3A+B.$$

Combine the coefficients of each matrix:

$$2A+3A-3B+B=5A-2B.$$

Now multiply each matrix by its scalar and subtract corresponding entries:

$$5A-2B = \begin{bmatrix}5&10\\-5&15\end{bmatrix} - \begin{bmatrix}8&-4\\0&10\end{bmatrix} = \begin{bmatrix}-3&14\\-5&5\end{bmatrix}.$$

So the simplified expression is

$$\boxed{\begin{bmatrix}-3&14\\-5&5\end{bmatrix}}.$$