Find the inverse of the following matrix using elementary operations.

A = \begin{bmatrix} 1 & 2 & -2\\ -1 & 3 & 0\\ 0 & -2 & 1 \end{bmatrix}

 

 

 

 
 
 
 
 

Answers (1)

A = \begin{bmatrix} 1 & 2 & -2\\ -1 & 3 & 0\\ 0 & -2 & 1 \end{bmatrix}     find A^{-1}

A = I A [using elementary operations]

I = A^{-1}A

\begin{bmatrix} 1 & 2 & -2\\ -1 & 3 & 0\\ 0 & -2 & 1 \end{bmatrix} = \begin{bmatrix} 1 &0 &0 \\ 0 & 1 &0 \\ 0 & 0 & 1 \end{bmatrix} A

A\qquad = \qquad I \quad \cdot \quad A

\begin{bmatrix} 1 & 2 & -2\\ 0 & 5 & -2\\ 0 & -2 & 1 \end{bmatrix} = \begin{bmatrix} 1 &0 &0 \\ 1 & 1 &0 \\ 0 & 0 & 1 \end{bmatrix} A \qquad R_2 \rightarrow R_2 + R_1

\begin{matrix} R_1 \rightarrow R_1 + 2R_3\\ R_2 \rightarrow R_2 + 2R_3 \end{matrix}    \begin{bmatrix} 1 & -2 & 0\\ 0 & 1 & 0\\ 0 & -2 & 1 \end{bmatrix} = \begin{bmatrix} 1 &0 &2 \\ 1 & 1 &2 \\ 0 & 0 & 1 \end{bmatrix} A

\begin{matrix} R_1 \rightarrow R_1 + 2R_2\\ R_3 \rightarrow R_3 + 2R_3 \end{matrix}    \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 3 &2 &6 \\ 1 & 1 &2 \\ 2 & 2 & 5 \end{bmatrix} A

\Rightarrow I = A^{-1}A

A^{-1}= \begin{bmatrix} 3 &2 &6 \\ 1 & 1 &2 \\ 2 & 2 & 5 \end{bmatrix}

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