Q36 / 90 · JEE Main 2017 (09 Apr Online) · Mathematics

For two $3 \times 3$ matrices $A$ and $B$ , let $A + B = 2 B^{'}$ and $3 A + 2 B = I_{3},$ where $B^{'}$ is the transpose of $B$ and $I_{3}$ is $3 \times 3$ identity matrix. Then :

Correct answer
A
$10 A + 5 B = 3 I_{3}$
B
$3 A + 6 B = 2 I_{3}$
C
$5 A + 10 B= 2 I_{3}$
D
$B + 2 A = I_{3}$

Solution

Given

A + B = 2 B^{T}

\Rightarrow A^{T} + B^{T} = 2 B

\Rightarrow B = \frac{A^{T} + B^{T}}{2}

= A + (\frac{B^{T} + A^{T}}{2}) = 2 B^{T}

2 A + A^{T} = 3 B^{T}

\Rightarrow A = \frac{3 B^{T} - A^{T}}{2}

3 A + 2 B = I_{3}

......

(i)

\Rightarrow 3 (\frac{3 B^{T} - A^{T}}{2}) + 2 (\frac{A^{T} + B^{T}}{2}) = I_{3}

\Rightarrow (\frac{3 B^{T} + 2 B^{T}}{2}) + 2 (\frac{A^{T} + B^{T}}{2}) = I_{3}

\Rightarrow 11 B^{T} - A^{T} = 2 I_{3}

.......

(i i)

Equation

(i) + (i i)

35 B = 7 I_{3}

\Rightarrow B = \frac{I_{3}}{5}

11 \frac{I_{3}}{5} - A = 2 I_{3}

\Rightarrow 11 \frac{I_{3}}{5} - 2 I_{3} = A

\Rightarrow A = \frac{I_{3}}{5}

∵ 5 A = 5 B = I_{3}

\Rightarrow 10 A + 5 B = 3 I_{3}

Hence, the correct answer is option: 'A'.

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