Matrices & Determinants
Characteristic Equation — Trace of A²
nta_pyq_2024_jan
Grade 12
Question:
Let $A$ be a $2\times2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $|A-xI|=0$ be $-1$ and $3$, then the sum of the diagonal elements of the matrix $A^2$ is
Step-by-Step Solution
Key Concept: The roots of $|A-xI|=0$ are eigenvalues: $-1$ and $3$. Sum of roots $=\text{tr}(A)=2$. Product $=\det(A)=-3$. $\text{tr}(A^2)=(\text{tr}A)^2-2\det A=4-2(-3)=10$.
Eigenvalues $-1,3$: $\text{tr}(A)=2$, $\det(A)=-3$. $\text{tr}(A^2)=(\text{tr}A)^2-2\det A=4+6=10$.
Correct Answer: 10