JEE Main 2022 · 25 Jun · Shift 2 · Mathematics Q21

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MathematicsInteger+4 / -1Question 21 of 90

Let $A=\begin{pmatrix}2&-2\\1&-1\end{pmatrix}$ and $B=\begin{pmatrix}-1&2\\-1&2\end{pmatrix}$ . Then the number of elements in the set $\{(n,m):n,m\in\{1,2,\dots,10\}$ and $nA^n+mB^m=I\}$ is ______.

Your answer

Answer and worked solution

Official answer: 1

Idempotent matrices. $A^2=A$ and $B^2=B$ (direct multiplication), so $A^n=A$ , $B^m=B$ .

Equation. $nA+mB=\begin{pmatrix}2n-m&-2n+2m\\n-m&-n+2m\end{pmatrix}=I$ needs $n=m$ and $2n-m=1$ : $n=m=1$ , which also gives the other entries.

One pair.

Answer: 1

Topics:Matrix algebra Special matrices

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JEE Main 2022 · 25 Jun · Shift 2, Mathematics question 21