Question

guys please help me with this question and whoever will give me the answer I will mark them as brainliest but I don't want useless answers and please give me step by step explanation​

Answer 1

Given :

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

$\rm I= \left[\begin{array}{ccc}1&0\\ \\ 0&1\end{array}\right]$

To find :

• A²-5A+7I

Solution :

To find A²-5A+7I , first we will find A².

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

$\implies\rm { A}^{2} = \left[\begin{array}{ccc}9 - 1&&3 - 2\\ \\ - 3 - 2&& - 1 + 4\end{array}\right]$

$\implies\rm { A}^{2} = \left[\begin{array}{ccc}8&&1\\ \\ - 5&& 3\end{array}\right]......(1)$

Now , we will find the value of 5A :-

$\implies\rm 5{ A} =5 \times \left[\begin{array}{ccc}3&1\\ \\ - 1&2\end{array}\right]$

$\implies\rm 5{ A} = \left[\begin{array}{ccc}3 \times 5&&1 \times 5\\ \\ - 1 \times 5&&2 \times 5\end{array}\right]$

$\implies\rm 5{ A} = \left[\begin{array}{ccc}15&& 5\\ \\ - 5&&10\end{array}\right]....(2)$

Now we will find 7I :-

$\implies\rm7 I= 7 \times \left[\begin{array}{ccc}1&0\\ \\ 0&1\end{array}\right]$

$\implies\rm7 I= \left[\begin{array}{ccc}7 \times1&&7 \times0\\ \\ 7 \times0&&7 \times1\end{array}\right]$

$\implies\rm7 I= \left[\begin{array}{ccc}7&0\\ \\ 0&7\end{array}\right] ........(3)$

Now , we will find the value of A²-5A+7I using (1),(2) and (3) :-

$\implies\rm { A}^{2} - 5 A + 7I= \left[\begin{array}{ccc}8&&1\\ \\ - 5&& 3\end{array}\right] - \left[\begin{array}{ccc}15&& 5\\ \\ - 5&&10\end{array}\right] + \left[\begin{array}{ccc}7&0\\ \\ 0&7\end{array}\right]$

$\implies\rm { A}^{2} - 5 A + 7I= \left[\begin{array}{ccc}8 - 15 + 7&&1 - 5 + 0\\ \\ - 5 + 5 + 0&& 3 - 10 + 7\end{array}\right]$

$\implies\rm { A}^{2} - 5 A + 7I= \left[\begin{array}{ccc}0&&- 4\\ \\ 0&& 0\end{array}\right]$

Answer :

$\bf { A}^{2} - 5 A + 7 \rm{I}= \left[\begin{array}{ccc}0&&- 4\\ \\ 0&& 0\end{array}\right]$