Let A - - 1 2- 3 4 - and B - - a b- c d - be two matrices such that they are commutative and c - 3b- Then- find the value of d - a-3b - c-

AB = #160;[ 1 amp;2; 3 amp; 4 ] #160;[ a amp; b; c amp; d ] = #160;[ a + 2c #160; amp; b + 2d; 3a + 4c a ...

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Byju's Answer

Standard XII

Mathematics

Scalar Triple Product

Let A = [ 1...

Question

Let $A = [\begin{matrix} 1 & 2 3 & 4 \end{matrix}]$ and $B = [\begin{matrix} a & b c & d \end{matrix}]$ be two matrices such that they are commutative and $c \neq 3 b$ . Then, find the value of $(d - a) / (3 b - c)$ .

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Solution

$A B = [\begin{matrix} 1 & 2 3 & 4 \end{matrix}] [\begin{matrix} a & b c & d \end{matrix}]$
$= [\begin{matrix} a + 2 c & b + 2 d 3 a + 4 c & 3 b + 4 d \end{matrix}]$
$B A = [\begin{matrix} a & b c & d \end{matrix}] [\begin{matrix} 1 & 2 3 & 4 \end{matrix}] = [\begin{matrix} a + 3 b & 2 a + 4 b c + 3 d & 2 c + 4 d \end{matrix}]$
If given matrices are commutative then $A B = B A$ ,
$\Rightarrow a + 2 c = a + 3 b$
$\Rightarrow 2 c = 3 b \Rightarrow b \neq 0$
$b + 2 d = 2 a + 4 b$
$\Rightarrow 2 a - 2 d = - 3 b$
$∴ \frac{d - a}{3 b - c} = \frac{\frac{3}{2} b}{3 b - \frac{3}{2} b} = 1$

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