For the matrices if then among the following which one is true? [2026]

Question

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]

Smart Achievers · Accepted Answer

(2)

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

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

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

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

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

$\Rightarrow 2 x - 11 y = 0$

Solution

Q.
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]

1 $x = 16, y = 3$

2 $x = 11, y = 2$

3 $x = 5, y = 7$

4 $x = 18, y = 11$

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Solution

Q. 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]

1 x=16, y=3

2 x=11, y=2

3 x=5, y=7

4 x=18, y=11