Answer:
A² - 6A + 11 I = [tex]\left[\begin{array}{ccc}0&0\\0&0\end{array}\right][/tex]
Step-by-step explanation:
Given the matrix
[tex]A=\left[\begin{array}{ccc}2&3\\-1&4\end{array}\right][/tex]
Calculate A² - 6A + 11 I
[tex]A^2 = A*A= \left[\begin{array}{ccc}2&3\\-1&4\end{array}\right] *\left[\begin{array}{ccc}2&3\\-1&4\end{array}\right] = \left[\begin{array}{ccc}2*2-3*1&2*3+3*4\\-1*2-4*1&-1*3+4*4\end{array}\right] =\left[\begin{array}{ccc}1&18\\-6&13\end{array}\right][/tex]
[tex]6A=6*\left[\begin{array}{ccc}2&3\\-1&4\end{array}\right] =\left[\begin{array}{ccc}12&18\\-6&24\end{array}\right][/tex]
[tex]11 I = 11 * \left[\begin{array}{ccc}1&0\\0&1\end{array}\right] =\left[\begin{array}{ccc}11&0\\0&11\end{array}\right][/tex]
∴ A² - 6A + 11 I = [tex]\left[\begin{array}{ccc}1&18\\-6&13\end{array}\right] -\left[\begin{array}{ccc}12&18\\-6&24\end{array}\right] +\left[\begin{array}{ccc}11&0\\0&11\end{array}\right] =\left[\begin{array}{ccc}0&0\\0&0\end{array}\right][/tex]
