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Algebra of Matrices — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices5 Marks
Question
If $\text{A}=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$ and $\text{B}=\begin{bmatrix}-1&3&5\\1&-3&-5\\-1&3&5\end{bmatrix},$ show that $AB = BA = O_{3\times 3}$
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Answer

Here,
$\text{AB}=\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}\begin{bmatrix}-1&3&5\\1&-3&-5\\-1&3&5\end{bmatrix}$
$\Rightarrow\text{AB}=\begin{bmatrix}-2-3+5&6+9-15&10+15-25\\1+4-5&-3-12+15&-5-20 +25\\-1-3+4&3+9-12&5+15-20\end{bmatrix}$
$\Rightarrow\text{AB}=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}$
$\Rightarrow\text{AB}=0_{3\times3}\ \dots(1)$
$\text{BA}=\begin{bmatrix}-1&3&5\\1&-3&-5\\-1&3&5\end{bmatrix}\begin{bmatrix}2&-3&-5\\-1&4&5\\1&-3&-4\end{bmatrix}$
$ \Rightarrow\text{BA}=\begin{bmatrix}-2-3+5&3+12-15&5+15-20\\2+3-5&-3-12+15&-5-15+20\\-2-3+5&3+12-15&5+15-20\end{bmatrix}$
$\Rightarrow\text{BA}=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}$
$\Rightarrow\text{BA}=0_{3\times3}\ \dots(2)$
$\Rightarrow\text{AB}=\text{BA}=0_{3\times3}$ [From eqs. (1) and (2)]

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