If A= [ cc 3 & -5 -4 & 2 ] find A^2-5 A-14 I Where I is unit matrix of order 2 2

Given & A= [ cc 3 & -5 -4 & 2 ] & A^2=A A= [ cc 3 & -5 -4 & 2 ] [ cc 3 & -5 -4 & 2 ] & = [ cc 9+20 & -15-10 -12-8 & 20+4 ] & = [ cc 29 & -25 -20 & 24 ] & 5 A=5 [ cc 3 & -5 -4 & 2 ] & = [ cc 15 & -25 -20 & 10 ] & A^2-5 A-14 I= [ cc 29 & -25 -20 & 24 ] & - [ cc 15 & -25 -20 & 10 ]-14 [ cc 1 & 0 0 & 1 ] & = [ cc 29 & -25 -20 & 24 ]- [ cc 15 & -20 -20 & 10 ]- [ cc 14 & 0 0 & 1 ] & = [ cc 29-15-14 & -25+25-0 -20+20+0 & 24-10-14 ] & = [ cc 0 & 0 0 & 0 ]

If A= [ cc 3 & -5 -4 & 2 ] find A^2-5 A-14 I Where I is unit matr…

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Question

If $A=\left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]$ find $A^2-5 A-14$ I
Where $I$ is unit matrix of order $2 \times 2$

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Answer

Given
$
\begin{aligned}
& A=\left[\begin{array}{cc}
3 & -5 \\
-4 & 2
\end{array}\right] \\
& A^2=A \times A=\left[\begin{array}{cc}
3 & -5 \\
-4 & 2
\end{array}\right]\left[\begin{array}{cc}
3 & -5 \\
-4 & 2
\end{array}\right] \\
& =\left[\begin{array}{cc}
9+20 & -15-10 \\
-12-8 & 20+4
\end{array}\right] \\
& =\left[\begin{array}{cc}
29 & -25 \\
-20 & 24
\end{array}\right] \\
& 5 A=5\left[\begin{array}{cc}
3 & -5 \\
-4 & 2
\end{array}\right] \\
& =\left[\begin{array}{cc}
15 & -25 \\
-20 & 10
\end{array}\right] \\
& \therefore A^2-5 A-14 I=\left[\begin{array}{cc}
29 & -25 \\
-20 & 24
\end{array}\right] \\
& -\left[\begin{array}{cc}
15 & -25 \\
-20 & 10
\end{array}\right]-14\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
29 & -25 \\
-20 & 24
\end{array}\right]-\left[\begin{array}{cc}
15 & -20 \\
-20 & 10
\end{array}\right]-\left[\begin{array}{cc}
14 & 0 \\
0 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
29-15-14 & -25+25-0 \\
-20+20+0 & 24-10-14
\end{array}\right] \\
& =\left[\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}\right]
\end{aligned}
$