The matrix A^2 + 4A - 5I, where I is identity matrix and A = [ * … - Vidyadip

MCQ

The matrix $A^2 + 4A - 5I$, where $I$ is identity matrix and $A = \left[ {\begin{array}{*{20}{c}}
1&2\\
4&{ - 3}
\end{array}} \right]$ , equals

✓
$4\left[ {\begin{array}{*{20}{c}}
2&1\\
2&0
\end{array}} \right]$
B
$4\left[ {\begin{array}{*{20}{c}}
0&{ - 1}\\
2&2
\end{array}} \right]$
C
$32\left[ {\begin{array}{*{20}{c}}
2&1\\
2&0
\end{array}} \right]$
D
$32\left[ {\begin{array}{*{20}{c}}
1&1\\
1&0
\end{array}} \right]$

✓

Answer

Correct option: A.

$4\left[ {\begin{array}{*{20}{c}}
2&1\\
2&0
\end{array}} \right]$

a
${A^2} + 4A - 51 = A \times A + 4A - 5I$

$ = \left[ {\begin{array}{*{20}{c}}
1&2\\
4&{ - 3}
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
1&2\\
4&{ - 3}
\end{array}} \right] + 4\left[ {\begin{array}{*{20}{c}}
1&2\\
4&{ - 3}
\end{array}} \right] - 5\left[ {\begin{array}{*{20}{c}}
1&0\\
0&1
\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}
9&{ - 4}\\
{ - 8}&{17}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
4&8\\
{16}&{ - 12}
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
5&0\\
0&5
\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}
{9 + 4 - 5}&{ - 4 + 8 - 0}\\
{ - 8 + 16 - 0}&{17 - 12 - 5}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
8&4\\
8&0
\end{array}} \right]$

$ = 4\left[ {\begin{array}{*{20}{c}}
2&1\\
2&0
\end{array}} \right]$

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