Consider the following matrices: 𝐴=[[-1,-3,1],[-3,-4,-4],[2,2,3]]𝐷=[5,0,2],[3,3,3],[0,-2,1]] Find a matrix 𝐵 such that 𝐴𝐵=𝐷 and a matrix 𝐶 such that 𝐶𝐴=𝐷

Given two matrices \( A \) and \( D \), we are tasked to find matrices \( B \) and \( C \) such that:

\[
AB = D \quad \text{and} \quad CA = D
\]

### The matrix \( A \) and \( D \) are defined as:

\[
A = \begin{bmatrix} -1 & -3 & 1 \\ -3 & -4 & -4 \\ 2 & 2 & 3 \end{bmatrix}
\]

\[
D = \begin{bmatrix} 5 & 0 & 2 \\ 3 & 3 & 3 \\ 0 & -2 & 1 \end{bmatrix}
\]

We will now solve the two matrix equations separately:

### 1. Finding matrix \( B \) such that \( AB = D \):

To find matrix \( B \), we need to solve the system:

\[
AB = D
\]

This can be rewritten as:

\[
B = A^{-1}D
\]

So, we first need to calculate the inverse of \( A \), \( A^{-1} \), and then multiply it by \( D \) to get matrix \( B \).

### 2. Finding matrix \( C \) such that \( CA = D \):

Similarly, to find matrix \( C \), we solve the system:

\[
CA = D
\]

This can be rewritten as:

\[
C = DA^{-1}
\]

Thus, we need to compute \( A^{-1} \) once and then use it for both \( B \) and \( C \).

I will now proceed with calculating \( A^{-1} \) and then find the matrices \( B \) and \( C \).### Results:

1. The inverse of matrix \( A \) is:
\[
A^{-1} = \begin{bmatrix}
-1.333 & 3.667 & 5.333 \\
0.333 & -1.667 & -2.333 \\
0.667 & -1.333 & -1.667
\end{bmatrix}
\]

2. The matrix \( B \) such that \( AB = D \) is:
\[
B = \begin{bmatrix}
4.333 & 0.333 & 13.667 \\
-3.333 & -0.333 & -6.667 \\
-0.667 & -0.667 & -4.333
\end{bmatrix}
\]

3. The matrix \( C \) such that \( CA = D \) is:
\[
C = \begin{bmatrix}
-5.333 & 15.667 & 23.333 \\
-1.000 & 2.000 & 4.000 \\
0.000 & 2.000 & 3.000
\end{bmatrix}
\]

Would you like more details on how these matrices were calculated or any additional clarifications?

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Here are five related questions for further exploration:

1. How do you calculate the inverse of a 3x3 matrix manually?
2. What are the conditions under which a matrix has an inverse?
3. Can you explain why matrix multiplication is not commutative?
4. What is the geometrical interpretation of matrix multiplication?
5. How do you solve a system of linear equations using matrices?

**Tip**: Always check if a matrix is invertible by ensuring its determinant is non-zero!

Consider the following matrices: A=[[-1,-3,1],[-3,-4,-4],[2,2,3]], D=[[5,0,2],[3,3,3],[0,-2,1]] Find a matrix B such that AB=D and a matrix C such that CA=D.

Solve for matrices B and C such that AB = D and CA = D, using the matrix inversion method. This problem demonstrates key concepts in linear algebra, including matrix multiplication and finding the inverse of a matrix. Suitable for college-level students.

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