Let A= 2 -2 1 -1 andB= -1 2 -1 2 . Then the number of elements in the set {n,m:n,m∈1,2,…….10 and nA^n+mB^m=I} is _____.

Correct answer: 1. Given, A= 2 -2 1 -1 So, A^2= 2 -2 1 -1 2 -2 1 -1 = 2 -2 1 -1 =A Also, B= -1 2 -1 2 B^2= -1 2 -1 2 -1 2 -1 2 = -1 2 -1 2 =B So, A+B= 2 -2 1 -1 + -1 2 -1 2 = 1 0 0 1 =I Therefore the equation nA^n+mB^m=I is true for n=1 & m=1 so, only one set is possible.

Q50 / 90 · JEE Main 2022 (25 Jun Shift 2) · Mathematics

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

Correct answer: 1

Solution

Given,

$A = (\begin{array}{l} 2 & - 2 \\ 1 & - 1 \end{array})$ So, $A^{2} = (\begin{array}{l} 2 & - 2 \\ 1 & - 1 \end{array}) (\begin{array}{l} 2 & - 2 \\ 1 & - 1 \end{array}) = (\begin{array}{l} 2 & - 2 \\ 1 & - 1 \end{array}) = A$

Also, $B = (\begin{matrix} - 1 & 2 \\ - 1 & 2 \end{matrix})$

$B^{2} = (\begin{matrix} - 1 & 2 \\ - 1 & 2 \end{matrix}) (\begin{matrix} - 1 & 2 \\ - 1 & 2 \end{matrix}) = (\begin{matrix} - 1 & 2 \\ - 1 & 2 \end{matrix}) = B$

So, $A + B = (\begin{array}{l} 2 & - 2 \\ 1 & - 1 \end{array}) + (\begin{matrix} - 1 & 2 \\ - 1 & 2 \end{matrix}) = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) = I$

Therefore the equation $n A^{n} + m B^{m} = I$ is true for $n = 1 & m = 1$ so, only one set is possible.

Hence, the correct answer is 1.

Let A=2-21-1 andB=-12-12 . Then the number of elements in the set {n,m:n,m∈1,2,…….10 and nAn+mBm=I} is _____.

Solution

Relevant Questions

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