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If \( A=\left[\begin{array}{ll}3 & -2 \\ \lambda & -2\end{array}\right] \), find the value of \( \lambda \) so that \( A^{2}=\lambda A + 4 I \).

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A = \(\begin{bmatrix}3&-2\\ \lambda&-2\end{bmatrix}\)

A2 = \(\begin{bmatrix}3&-2\\ \lambda&-2\end{bmatrix}\) \(\begin{bmatrix}3&-2\\ \lambda&-2\end{bmatrix}\) = \(\begin{bmatrix}9-2\lambda&-2\\ \lambda&-2\lambda+4\end{bmatrix}\)

\(\begin{bmatrix}9-2\lambda&-2\\ \lambda&-2\lambda+4\end{bmatrix}\) = \(\begin{bmatrix}9-2\lambda&-2\\ \lambda&-2\lambda+4\end{bmatrix}\) 

\(\therefore\) -2\(\lambda\) = -2

⇒ \(\lambda=1\)

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