JEE Main 2022 · 25 Jun · Shift 1 · Mathematics Q11

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MathematicsSingle correct+4 / -1Question 11 of 90

Let $A=\begin{bmatrix}0&-2\\2&0\end{bmatrix}$ . If M and N are two matrices given by $M=\displaystyle\sum_{k=1}^{10}A^{2k}$ and $N=\displaystyle\sum_{k=1}^{10}A^{2k-1}$ then $MN^2$ is :

a non-identity symmetric matrix

a skew-symmetric matrix

neither symmetric nor skew-symmetric matrix

an identity matrix

Answer and worked solution

Official answer: A

Powers. $A^2=-4I$ , so $A^{2k}=(-4)^kI$ and $A^{2k-1}=(-4)^{k-1}A$ .

So $M=mI$ and $N=nA$ with scalars $m=\sum(-4)^k\ne0$ , $n=\sum(-4)^{k-1}$ ; $N^2=n^2A^2=-4n^2I$ .

Product. $MN^2=-4mn^2I$ , a scalar matrix: symmetric, and not $I$ (its entry is a large integer).

(A)

Topics:Matrix algebra Special matrices

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JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 11