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If \( A=\left[\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right] \), find the values of \( a \) and \( b \) such that \( A^{2}+a A+b \mid=0 \) |
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Answer» A = \(\begin{bmatrix}3&2\\1&1\end{bmatrix}\) Then A2 = \(\begin{bmatrix}3&2\\1&1\end{bmatrix}\)\(\begin{bmatrix}3&2\\1&1\end{bmatrix}\) = \(\begin{bmatrix}3\times3+2\times1&3\times2+2\times1\\1\times3+1\times1&1\times2+1\times1\end{bmatrix}\) = \(\begin{bmatrix}11&8\\4&3\end{bmatrix}\) Given that A2 + aA + bI = 0 \(\therefore\) \(\begin{bmatrix}11&8\\4&3\end{bmatrix}\) + a\(\begin{bmatrix}3&2\\1&1\end{bmatrix}\) + b\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\) = \(\begin{bmatrix}0&0\\0&0\end{bmatrix}\) ⇒ \(\begin{bmatrix}11+3a+b&8+2a\\4+a&3+a+b\end{bmatrix}\) = \(\begin{bmatrix}0&0\\0&0\end{bmatrix}\) \(\therefore\) 4 + a = 0 (By comparing a21 element of both equal matrices) ⇒ a = -4 And 3 + a + b = 0 ⇒ b = -(a + 3) = -(-4 + 3) = -(-1) = 1 \(\therefore\) a = -4 and b = 1 |
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