Matrices and Determinants jee mains questions | JEE Main Maths Matrices and Determinants Previous Year Question

Q 11 :

For the matrices $A = [[\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]] and B = [[\begin{matrix} - 29 & 49 \\ - 13 & 18 \end{matrix}]],$ if $((A^{15} + B)) [[\begin{matrix} x \\ y \end{matrix}]] = [[\begin{matrix} 0 \\ 0 \end{matrix}]],$ then among the following which one is true? [2026]

$x = 16, y = 3$
$x = 11, y = 2$
$x = 5, y = 7$
$x = 18, y = 11$

(2)

$Here A^{n} = [\begin{matrix} 2 n + 1 & - 4 n \\ n & - 2 n + 1 \end{matrix}]$

$\Rightarrow A^{15} = [\begin{matrix} 31 & - 60 \\ 15 & - 29 \end{matrix}]$

$\Rightarrow A^{15} + B = [\begin{matrix} 2 & - 11 \\ 2 & - 11 \end{matrix}]$

$Now (A^{15} + B) [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]$

$\Rightarrow [\begin{matrix} 2 & - 11 \\ 2 & - 11 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]$

$\Rightarrow 2 x - 11 y = 0$

Q 12 :

Let A, B and C be three $2 \times 2$ matrices with real entries such that $B = {(I + A)}^{- 1} and A + C = I .$ If $B C = [[\begin{matrix} 1 & - 5 \\ - 1 & 2 \end{matrix}]] and C B [[\begin{matrix} x_{1} \\ x_{2} \end{matrix}]] = [[\begin{matrix} 12 \\ - 6 \end{matrix}]],$ then $x_{1} + x_{2}$ is. [2026]

-2
4
2
0

(4)

$B = {(I + A)}^{- 1}, A + C = I$

$\Rightarrow B (I + A) = (I + A) B = I$

$\Rightarrow B + B A = B + A B$

$\Rightarrow B + B (I - C) = B + (I - C) B$

$\Rightarrow 2 B - B C = 2 B - C B$

$\Rightarrow B C = C B$

$∴$ $C B [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 1 & - 5 \\ - 1 & 2 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 12 \\ - 6 \end{matrix}]$

$\Rightarrow [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = {[\begin{matrix} 1 & - 5 \\ - 1 & 2 \end{matrix}]}^{- 1} [\begin{matrix} 12 \\ - 6 \end{matrix}] = \frac{- 1}{3} [\begin{matrix} 2 & 5 \\ 1 & 1 \end{matrix}] [\begin{matrix} 32 \\ - 6 \end{matrix}]$

$\Rightarrow [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 2 \\ - 2 \end{matrix}]$

$∴$ $x_{1} + x_{2} = 0$

Topic Question Set

Q 11 :

For the matrices $A = [[\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]] and B = [[\begin{matrix} - 29 & 49 \\ - 13 & 18 \end{matrix}]],$ if $((A^{15} + B)) [[\begin{matrix} x \\ y \end{matrix}]] = [[\begin{matrix} 0 \\ 0 \end{matrix}]],$ then among the following which one is true? [2026]

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Topic Question Set

Q 11 : For the matrices A=[3-41-1] and B=[-2949-1318], if (A15+B)[xy]=[00], then among the following which one is true? [2026]

Q 12 : Let A, B and C be three 2×2 matrices with real entries such that B=(I+A)-1 and A+C=I. If BC=[1-5-12] and CB[x1x2]=[12-6], then x1+x2 is. [2026]

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Q 11 :

For the matrices $A = [[\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]] and B = [[\begin{matrix} - 29 & 49 \\ - 13 & 18 \end{matrix}]],$ if $((A^{15} + B)) [[\begin{matrix} x \\ y \end{matrix}]] = [[\begin{matrix} 0 \\ 0 \end{matrix}]],$ then among the following which one is true? [2026]