https://file.aiquickdraw.com/mathbot/file/1726265474593zwn7anmm.jpg

The matrix equation given in the image is:

\[
\begin{bmatrix} 4 \\ 1 \end{bmatrix} X = \begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}
\]

This represents a system where we want to find a 2×2 matrix \( X \) such that when we multiply it with the column vector \(\begin{bmatrix} 4 \\ 1 \end{bmatrix}\), the result is the matrix on the right-hand side.

We can solve this by setting up the system of equations that result from matrix multiplication.

Let \( X = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).

The matrix multiplication \(\begin{bmatrix} 4 \\ 1 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) would give us:

\[
4a + c \quad 4b + d
\]

Matching this with the right-hand side, \(\begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}\), gives us a system of equations:

1. \( 4a + c = -4 \)
2. \( 4b + d = 8 \)
3. \( a + c = -1 \)
4. \( b + d = 2 \)

Now, we can solve this system to find \(a, b, c, d\).

### Step 1: Solve for \(c\) and \(d\)
From equation 3: \( a + c = -1 \), we can express \(c\) as:
\[
c = -1 - a
\]

From equation 4: \( b + d = 2 \), we can express \(d\) as:
\[
d = 2 - b
\]

### Step 2: Substitute into equations 1 and 2
Now, substitute \( c = -1 - a \) into equation 1:
\[
4a + (-1 - a) = -4
\]
Simplifying:
\[
4a - 1 - a = -4 \implies 3a = -3 \implies a = -1
\]

Now substitute \( d = 2 - b \) into equation 2:
\[
4b + (2 - b) = 8
\]
Simplifying:
\[
4b + 2 - b = 8 \implies 3b = 6 \implies b = 2
\]

### Step 3: Find \(c\) and \(d\)
Now, substitute \( a = -1 \) into \( c = -1 - a \):
\[
c = -1 - (-1) = 0
\]

Substitute \( b = 2 \) into \( d = 2 - b \):
\[
d = 2 - 2 = 0
\]

### Final Result:
The matrix \( X \) is:
\[
X = \begin{bmatrix} -1 & 2 \\ 0 & 0 \end{bmatrix}
\]

Let me know if you would like further details or have any questions.

Here are some related questions to explore further:
1. How do we multiply matrices?
2. Can matrix multiplication be performed in any order?
3. What is the role of the identity matrix in matrix equations?
4. How can we solve matrix equations with larger matrices?
5. What are some real-world applications of matrix equations?

**Tip:** When solving matrix equations, always carefully match the dimensions of matrices involved to ensure the operations are valid.

Solve the matrix equation [4 1]^T * X = [-4 8; -1 2]

This page walks through solving the matrix equation [4 1]^T * X = [-4 8; -1 2], breaking it down into a system of linear equations. Learn step-by-step how to solve for X by identifying key matrix multiplication concepts. Suitable for Grades 10-12.

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