If A= \begin{bmatrix} 1 &-1 &1 \\2 & -1 & 0\\1 &0 &0 \end{bmatrix}, find A^2 and show that A^2=A^{-1}.

 

 

 

 
 
 
 
 

Answers (1)

Let A= \begin{bmatrix} 1 &-1 &1 \\2 & -1 & 0\\1 &0 &0 \end{bmatrix} A^2 and prove that A^2=A^{-1}

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

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

A=IA

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

R_2\rightarrow R_2-2R_1

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

R_3\rightarrow R_3-R_1

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

R_1\rightarrow R_1+R_2

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

R_3\rightarrow R_3-R_2

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

R_2\rightarrow R_2+2R_3

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

R_1\rightarrow R_1+R_3

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

I=A^{-1}A

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

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