JEE Main 2021 · 24 Feb · Shift 1 · Mathematics Q83

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MathematicsInteger+4 / -1Question 83 of 90

Let $P=\begin{bmatrix}3&-1&-2\\2&0&\alpha\\3&-5&0\end{bmatrix}$ , where α ∈ ℝ. Suppose Q = [q $_{ij}$ ] is a matrix satisfying PQ = kI $_3$ for some non-zero k ∈ ℝ. If $q_{23}=-\dfrac k8$ and $|Q|=\dfrac{k^2}2$ , then α $^2$ + k $^2$ is equal to _______.

Your answer

Answer and worked solution

Official answer: 17

Q = kP⁻¹. Then $|Q|=\frac{k^3}{|P|}=\frac{k^2}2$ gives $|P|=2k$ , and $|P|=12\alpha+20$ .

Entry. $q_{23}=\frac{k}{|P|}C_{32}=\frac{-(3\alpha+4)}2=-\frac k8$ , so $k=4(3\alpha+4)$ . With $k=6\alpha+10$ : $\alpha=-1$ , $k=4$ .

Answer: 17

Topics:Adjoint and inverse Determinants and properties

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JEE Main 2021 · 24 Feb · Shift 1, Mathematics question 83