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The problem asks us to find the matrix \( A \), such that:

\[
A \begin{bmatrix} 1 & 0 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} -4 & -3 \\ -1 & 15 \end{bmatrix}
\]

We are given the hint that matrix \( A \) has the form:

\[
A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}
\]

We need to determine the values of \( a \), \( b \), \( c \), and \( d \).

### Step 1: Matrix Multiplication Setup

We will multiply matrix \( A \) by the matrix on the left-hand side:

\[
\begin{bmatrix} a & b \\ c & d \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ -1 & 3 \end{bmatrix}
\]

The result of this multiplication is:

\[
= \begin{bmatrix} a(1) + b(-1) & a(0) + b(3) \\ c(1) + d(-1) & c(0) + d(3) \end{bmatrix}
= \begin{bmatrix} a - b & 3b \\ c - d & 3d \end{bmatrix}
\]

### Step 2: Set Equal to the Given Matrix

We now set this result equal to the right-hand side matrix:

\[
\begin{bmatrix} a - b & 3b \\ c - d & 3d \end{bmatrix} = \begin{bmatrix} -4 & -3 \\ -1 & 15 \end{bmatrix}
\]

### Step 3: Solve the System of Equations

This gives us a system of equations by comparing corresponding elements:

1. \( a - b = -4 \)
2. \( 3b = -3 \)
3. \( c - d = -1 \)
4. \( 3d = 15 \)

### Step 4: Solve for \( b \) and \( d \)

From equation (2): 
\[
3b = -3 \implies b = -1
\]

From equation (4): 
\[
3d = 15 \implies d = 5
\]

### Step 5: Solve for \( a \) and \( c \)

Substitute \( b = -1 \) into equation (1):
\[
a - (-1) = -4 \implies a + 1 = -4 \implies a = -5
\]

Substitute \( d = 5 \) into equation (3):
\[
c - 5 = -1 \implies c = 4
\]

### Final Solution

Thus, the matrix \( A \) is:

\[
A = \begin{bmatrix} -5 & -1 \\ 4 & 5 \end{bmatrix}
\]

Would you like further details or have any questions?

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Here are 5 related questions:
1. How would you solve this if matrix \( A \) was a 3x3 matrix instead?
2. What is the significance of matrix multiplication in this context?
3. Can you find the inverse of the matrix \( A \) we calculated?
4. How would you solve this system using determinants?
5. How do these types of problems relate to systems of linear equations?

**Tip:** When multiplying matrices, always keep the order of multiplication in mind—matrix multiplication is not commutative!

Find the matrix A such that A * [[1, 0], [-1, 3]] = [[-4, -3], [-1, 15]].

Learn how to solve a matrix multiplication problem by finding matrix A such that A * [[1, 0], [-1, 3]] equals [[-4, -3], [-1, 15]]. This example covers matrix multiplication, linear algebra concepts, and solving systems of equations, suitable for advanced high school or college-level students.

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