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(C) The equation $|A-xI|=0$ is the characteristic equation of matrix $A$.Given the roots are $\lambda_1 = -1$ and $\lambda_2 = 3$.The sum of the roots

Vedclass · Accepted Answer

$10$

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(C) The equation $|A-xI|=0$ is the characteristic equation of matrix $A$.Given the roots are $\lambda_1 = -1$ and $\lambda_2 = 3$.The sum of the roots (trace of $A$) is $\operatorname{tr}(A) = \lambda_1 + \lambda_2 = -1 + 3 = 2$.The product of the roots (determinant of $A$) is $|A| = \lambda_1 \lambda_2 = (-1)(3) = -3$.By the Cayley-Hamilton theorem,every square matrix satisfies its own characteristic equation: $A^2 - \operatorname{tr}(A)A + |A|I = 0$.Substituting the values: $A^2 - 2A - 3I = 0$,which implies $A^2 = 2A + 3I$.Taking the trace on both sides: $\operatorname{tr}(A^2) = \operatorname{tr}(2A + 3I) = 2\operatorname{tr}(A) + 3\operatorname{tr}(I)$.Since $\operatorname{tr}(A) = 2$ and $\operatorname{tr}(I) = 1 + 1 = 2$,we have:$\operatorname{tr}(A^2) = 2(2) + 3(2) = 4 + 6 = 10$.

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

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