Question
Reduce the Matrix
to a diagonal form.
Answer :
Word Count : 470
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To reduce the given matrix \( M \) to a diagonal form, we can perform a series of row and column operations. Here's a step-by-step solution: ### Given Matrix: \[ M = \begin{bmatrix} 1 & 2 & -2 \\ 2 & 1 & 2 \\ -2 & 2 & 1 \end{bmatrix} \] ### Step 1: Eliminate the off-diagonal elements in the first column. - Row 2 = Row 2 - 2 × Row 1 \[ \begin{bmatrix} 2 & 1 & 2 \end{bmatrix} - 2 \times \begin{bmatrix} 1 & 2 & -2 \end{bmatrix} = \begin{bmatrix} 0 & -3 & 6 \end{bmatrix} \] - Row 3 = Row 3 + 2 × Row 1 \[ \begin{bmatrix} -2 & 2 & 1 \end{bmatrix} + 2 \times \begin{bmatrix} 1 & 2 & -2 \end{bmatrix} = \begin{bmatrix} 0 & 6 & -3 \end{bmatrix} \] After these operations, the matrix becomes: \[ \begin{bmatrix} 1 & 2 & -2 \\ 0 & -3 & 6 \\ 0 & 6 & -3 \end{bmatrix} \] ### Step 2: Eliminate the off-diagonal elements in
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