Question

please solve in detail it's very important and urgent please​

Answer 1

SOLUTION

$A\: = \: \left[\begin{array}{ccc}2&0&1\\2&1&-3\\1&-1&0\end{array}\right]$

$A^2\: = \: \left[\begin{array}{ccc}2&0&1\\2&1&-3\\1&-1&0\end{array}\right]\: \left[\begin{array}{ccc}2&0&1\\2&1&-3\\1&-1&0\end{array}\right]$

$A^2\: =\:\left[\begin{array}{ccc}2\times2+0\times2+1\times1&2\times0+0\times1+1\times(-1)&2\times1+0\times(-3)+1\times0\\2\times2+1\times2+(-3)\times(1)&2\times0+1\times1+(-3)\times(-1)&2\times1+1\times(-3)+(-3)\times0\\1\times2+(-1)\times2+0\times1&1\times0+(-1)\times(1)+0\times(-1)&1\times(1)+(-1)\times(-3)+0\times0\end{array}\right]$

$A^2\: =\: \left[\begin{array}{ccc}4+0+1&0+0-1&2+0+0\\4+2-3&0+1+3&2-3+0\\2-2+0&0-1-0)&1+3+0\end{array}\right]$

$A^2\: = \: \left[\begin{array}{ccc}5&-1&2\\3&4&-1\\0&-1&4\end{array}\right]$

Now,

f(x) = x²-5x+6

f(A) = A² - 5A + 6I

$f(A)\: = \: \left[\begin{array}{ccc}5&-1&2\\3&4&-1\\0&-1&4\end{array}\right]\: - \:5\:\left[\begin{array}{ccc}2&0&1\\2&1&-3\\1&-1&0\end{array}\right]\: +\:\left[\begin{array}{ccc}6&0&0\\0&6&0\\0&0&6\end{array}\right]$

$f(A)\: = \: \left[\begin{array}{ccc}5&-1&2\\3&4&-1\\0&-1&4\end{array}\right]\: - \:\left[\begin{array}{ccc}10&0&5\\10&5&-15\\5&-5&0\end{array}\right]\: +\:\left[\begin{array}{ccc}6&0&0\\0&6&0\\0&0&6\end{array}\right]$

$f(A)\: = \: \left[\begin{array}{ccc}5-10+6&-1-0+0&2-5+0\\3-10+0&4-5+6&-1+15+0\\0-5+0&-1+5+0&4-0+6\end{array}\right]$

$f(A) \: = \: \left[\begin{array}{ccc}1&-1&-3\\-7&5&14\\-5&4&10\end{array}\right]$