Question
Answer :
Word Count : 671
We are given the matrix: \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix} \] and the identity matrix: \[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] The inverse of \( A \) is given by: \[ A^{-1} = \frac{1}{6} (A^2 + aA + bI) \] ### Step 1: Compute \( A^2 \) First, compute \( A^2 = A \cdot A \): \[ A^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix} \] Compute each element: - First row: - \( (1,1) \): \( 1 \times 1 + 0 \times 0 + 0 \times 0 = 1 \) - \( (1,2) \): \( 1 \times 0 + _________ ____ __________ __________ ___ _________ _________ ________ _____ _________ ___ _______.
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We are given the matrix: \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix} \] and the identity matrix: \[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] The inverse of \( A \) is given by: \[ A^{-1} = \frac{1}{6} (A^2 + aA + bI) \] ### Step 1: Compute \( A^2 \) First, compute \( A^2 = A \cdot A \): \[ A^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \end{bmatrix} \] Compute each element: - First row: - \( (1,1) \): \( 1 \times 1 + 0 \times 0 + 0 \times 0 = 1 \) - \( (1,2) \): \( 1 \times 0 + _________ ____ __________ __________ ___ _________ _________ ________ _____ _________ ___ _______.
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