The question asks us to solve for $x$ and $y$ given the matrix equation:

$\begin{bmatrix} 7 & 2 \\ -4 & 23 \end{bmatrix} = x \begin{bmatrix} 3 & -1 \\ 2 & -5 \end{bmatrix} + y \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

This can be broken down into a system of linear equations by comparing the entries of the resulting matrices on both sides.

Step 1: Expand the Right Side

1. Multiply $x$ by each entry of the matrix $\begin{bmatrix} 3 & -1 \\ 2 & -5 \end{bmatrix}$. $x \begin{bmatrix} 3 & -1 \\ 2 & -5 \end{bmatrix} = \begin{bmatrix} 3x & -x \\ 2x & -5x \end{bmatrix}$

2. Multiply $y$ by each entry of the identity matrix $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. $y \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} y & 0 \\ 0 & y \end{bmatrix}$

3. Add the matrices: $\begin{bmatrix} 3x + y & -x \\ 2x & -5x + y \end{bmatrix}$

Step 2: Set Up Equations by Comparing Entries

Now, equate each element in the resulting matrix to the corresponding element in $\begin{bmatrix} 7 & 2 \\ -4 & 23 \end{bmatrix}$:

1. Top-left element: $3x + y = 7$
2. Top-right element: $-x = 2$
3. Bottom-left element: $2x = -4$
4. Bottom-right element: $-5x + y = 23$

Step 3: Solve for $x$ and $y$

From equation $2x = -4$: $x = -2$

Substitute $x = -2$ into the other equations:

1. Substitute $x = -2$ into $-x = 2$, which holds true.

2. Substitute $x = -2$ into $3x + y = 7$: $3(-2) + y = 7 \Rightarrow -6 + y = 7 \Rightarrow y = 13$

3. Substitute $x = -2$ into $-5x + y = 23$: $-5(-2) + y = 23 \Rightarrow 10 + y = 23 \Rightarrow y = 13$

Solution

The values of $x$ and $y$ are: $x = -2 \quad \text{and} \quad y = 13$

Answer

The correct answer is B. -2 dan 13.

Would you like further details on any step?

Here are some related questions to consider:

1. What are the basic properties of matrix addition and scalar multiplication?
2. How can we verify solutions in matrix equations?
3. What are some common techniques for solving systems of linear equations?
4. How does matrix multiplication differ from scalar multiplication?
5. Can matrix equations have multiple solutions or be inconsistent?

Tip: Always check each substituted value in all initial equations to verify your solution.