Let A be a [2 2] real matrix and I be the identity matrix of order 2.

Question

Let $A$ be a $$2 \times 2$$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $$|\mathrm{A}-x \mathrm{I}|=0$$ be $-1$ and 3, then the sum of the diagonal elements of the matrix $$\mathrm{A}^2$$ is

Answer: 10

Accepted Answer

$|A-xI|=0$

Roots are $-$1 and 3

Sum of roots $=tr(A)=2$

Product of roots $=|A|=-3$

Let $A=[\begin{gathered} a b \ c d ]$

We have $a+d=2$

$ad-bc=-3$

${A}^{2}=[ a b \ c d ]	imes [ a b \ c d ]=[ {a}^{2}+bc ab+bd \ ac+cd bc+{d}^{2} \end{gathered}]$

We need ${a}^{2}+bc+bc+{d}^{2}$

$\begin{gathered} ={a}^{2}+2bc+{d}^{2} \ =(a+d{)}^{2}-2ad+2bc \ =4-2(ad-bc) \ =4-2(-3) \ =4+6 \ =10 \end{gathered}$

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