If A=\begin{bmatrix} 4 & 2 \\ -1& 1 \end{bmatrix}, show that \left ( A-2I \right )(A-3I)=0.

 

 

 

 
 
 
 
 

Answers (1)

A=\begin{bmatrix} 4 &2 \\ -1&1 \end{bmatrix}  and I=\begin{bmatrix} 1 &0 \\ 0&1 \end{bmatrix}

A-2I=\begin{bmatrix} 4 &2 \\ -1&1 \end{bmatrix}-\begin{bmatrix} 2 &0 \\ 0&2 \end{bmatrix}

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

A-3I=\begin{bmatrix} 4 &2 \\ -1 &1 \end{bmatrix}-\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}

                \Rightarrow \begin{bmatrix} 1 &2 \\ -1&-2 \end{bmatrix}

Then \left ( A-2I \right )(A-3I)=0

\begin{bmatrix} 2 &2 \\ -1&-1 \end{bmatrix}*\begin{bmatrix} 1 &2 \\ -1&-2 \end{bmatrix}\Rightarrow \begin{bmatrix} 0 &0 \\ 0 & 0 \end{bmatrix}

                                                 =0

Hence proved.

 

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