JEE Advanced 2020 · Paper 2 · Mathematics Q4

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MathematicsInteger+3 / -1Question 40 of 54

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2\times2$ matrix such that the trace of $A$ is 3 and the trace of $A^3$ is $-18$ , then the value of the determinant of $A$ is ______

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Answer and worked solution

Official answer: 5

Let the eigenvalues of $A$ be $\lambda_1,\lambda_2$ , so that $\lambda_1+\lambda_2=3$ and $\lambda_1\lambda_2=\det A$ . Then
$\operatorname{tr}A^3=\lambda_1^3+\lambda_2^3=(\lambda_1+\lambda_2)^3-3\lambda_1\lambda_2(\lambda_1+\lambda_2)=27-9\det A=-18.$
So $\det A=5$ .

Answer: 5

Topics:Matrix algebra Roots and coefficients

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JEE Advanced 2020 · Paper 2, Mathematics question 4