Question 22. 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 ____
Answer (10)
Explanation:
|A-x I|=0Roots 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}{lcc}a&b\;c&d\end{array}\right]\times\left[\begin{array}{lcc}a&b\;c&d\end{array}\right] =\left[\begin{array}{lcc}a^2+bc&ab+bd\;ac+cd&bc+d^2\end{array}\right]We need a^{2}+bc+bc+d^{2}
=a^2+2bc+d^2 =\left(a+d\right)^2-2ad+2bc =4-2\left(ad-bc\right) =4-2\left(-3\right)= 4 + 6 = 10