Let's break down and solve the matrix expression step by step.

The given expression is:

$5 \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix} - 2 \begin{bmatrix} 5 & 0 \\ -6 & 7 \end{bmatrix} + 5 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Step 1: Multiply the scalar by each matrix

First matrix:

$5 \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix} = \begin{bmatrix} 5(1) & 5(2) \\ 5(3) & 5(-4) \end{bmatrix} = \begin{bmatrix} 5 & 10 \\ 15 & -20 \end{bmatrix}$

Second matrix:

$-2 \begin{bmatrix} 5 & 0 \\ -6 & 7 \end{bmatrix} = \begin{bmatrix} -2(5) & -2(0) \\ -2(-6) & -2(7) \end{bmatrix} = \begin{bmatrix} -10 & 0 \\ 12 & -14 \end{bmatrix}$

Third matrix:

$5 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5(1) & 5(0) \\ 5(0) & 5(1) \end{bmatrix} = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$

Step 2: Add/Subtract the matrices

Now we need to combine the results from Step 1:

$\begin{bmatrix} 5 & 10 \\ 15 & -20 \end{bmatrix} + \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} - \begin{bmatrix} -10 & 0 \\ 12 & -14 \end{bmatrix}$

First, subtract the second matrix: $\begin{bmatrix} 5 & 10 \\ 15 & -20 \end{bmatrix} - \begin{bmatrix} -10 & 0 \\ 12 & -14 \end{bmatrix} = \begin{bmatrix} 5 - (-10) & 10 - 0 \\ 15 - 12 & -20 - (-14) \end{bmatrix} = \begin{bmatrix} 15 & 10 \\ 3 & -6 \end{bmatrix}$

Then add the third matrix: $\begin{bmatrix} 15 & 10 \\ 3 & -6 \end{bmatrix} + \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} = \begin{bmatrix} 15 + 5 & 10 + 0 \\ 3 + 0 & -6 + 5 \end{bmatrix} = \begin{bmatrix} 20 & 10 \\ 3 & -1 \end{bmatrix}$

Final Answer:

The result of the matrix operation is:

$\begin{bmatrix} 20 & 10 \\ 3 & -1 \end{bmatrix}$

Let me know if you'd like more details on the steps or have further questions!

Related questions:

1. How do you perform scalar multiplication with matrices?
2. What is the process for matrix addition and subtraction?
3. Can matrix multiplication be performed on two non-square matrices?
4. What are some common properties of matrix operations?
5. How do identity matrices affect matrix operations?

Tip:

Always check matrix dimensions before performing any operations to ensure they're compatible for addition, subtraction, or multiplication.