Matrices Question 4
Question 4 - 2024 (27 Jan Shift 2)
Let A be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2 . If the roots of the equation $|A-x I|=0$ be -1 and 3 , then the sum of the diagonal elements of the matrix $A^{2}$ is.
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Answer (10)
Solution
$|A-x I|=0$
Roots are -1 and 3
Sum of roots $=\operatorname{tr}(A)=2$
Product of roots $=|A|=-3$
Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$
We have $a+d=2$
$ad-bc=-3$
$A^{2}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \times\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left[\begin{array}{ll}a^{2}+b c & a b+b d \\ a c+c d & b c+d^{2}\end{array}\right]$
We need $a^{2}+bc+bc+d^{2}$
$=a^{2}+2 b c+d^{2}$
$=(a+d)^{2}-2 a d+2 b c$
$=4-2(a d-b c)$
$=4-2(-3)$
$=4+6$
$=10$
