Let's solve the matrix equation step by step: Given: \[ \begin{bmatrix} 1 & 2 & 3 \\ 3 & 7 & 6 \\ 1 & 0 & 8 \end{bmatrix} X = \begin{bmatrix} 1 & 4 & -2 & 0 & 3 \\ 0 & -1 & 5 & 2 & 7 \\ -3 & 6 & 8 & 9 & 0 \end{bmatrix} \] ### **Step 1: Write the system as \( AX = B \)** Here, - \(A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 7 & 6 \\ 1 & 0 & 8 \end{bmatrix}\) - \(X\) is the unknown matrix. - \(B = \begin{bmatrix} 1 & 4 & -2 & 0 & 3 \\ 0 & -1 & 5 & 2 & 7 \\ -3 & 6 & 8 & 9 & 0 \end{bmatrix}\) **Explanation:** We recognize this as a standard matrix equation of the form \( AX = B \). --- ### **Step 2: Express \( X \) in terms of \( A \) and \( B \)** To solve for \( X \), multiply both sides by \( A^{-1} \) (the inverse of \(A\)): \[ A X = B \implies X = A^{-1} B \] **Explanation:** We use the property that multiplying both sides of a matrix equation by the inverse of the coefficient matrix isolates the unknown matrix \(X\). --- ## **Final Answer:** \[ \boxed{ X = A^{-1} B = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 7 & 6 \\ 1 & 0 & 8 \end{bmatrix}^{-1} \begin{bmatrix} 1 & 4 & -2 & 0 & 3 \\ 0 & -1 & 5 & 2 & 7 \\ -3 & 6 & 8 & 9 & 0 \end{bmatrix} } \] This is the solution for \( X \) in terms of \( A \) and \( B \). If you need the explicit numeric solution, you would have to compute \( A^{-1} \) and then multiply it by \( B \).