Questions: The inverse of matrix (A) is given. [ A^-1 = left[ beginarraylll 3 2 1 1 1 2 1 2 1 endarray right] ] Use the inverse to solve for (X). [ AX = left[ beginarrayr 3 -1 2 endarray right] . ] [ X = left[ beginarrayc endarray right] ]

Solution Steps

To solve for \( X \) in the equation \( AX = B \) using the inverse of matrix \( A \), we can use the property that \( X = A^{-1}B \). Given \( A^{-1} \) and \( B \), we can compute \( X \) by multiplying \( A^{-1} \) with \( B \).

We are given the inverse of matrix \( A \) as follows: \[ A^{-1} = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 1 & 2 \\ 1 & 2 & 1 \end{bmatrix} \] and the matrix \( B \): \[ B = \begin{bmatrix} 3 \\ -1 \\ 2 \end{bmatrix} \]

To find \( X \), we use the formula: \[ X = A^{-1}B \] Calculating this gives: \[ X = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 1 & 2 \\ 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ -1 \\ 2 \end{bmatrix} \]

Calculating each element of \( X \):

• First element: \[ 3 \cdot 3 + 2 \cdot (-1) + 1 \cdot 2 = 9 - 2 + 2 = 9 \]
• Second element: \[ 1 \cdot 3 + 1 \cdot (-1) + 2 \cdot 2 = 3 - 1 + 4 = 6 \]
• Third element: \[ 1 \cdot 3 + 2 \cdot (-1) + 1 \cdot 2 = 3 - 2 + 2 = 3 \]

Thus, we find: \[ X = \begin{bmatrix} 9 \\ 6 \\ 3 \end{bmatrix} \]

Final Answer