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Algebra of Matrices — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSAlgebra of Matrices3 Marks
Question
Let $\text{A}=\begin{bmatrix}2&-3\\-7&5\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0\\2&-4\end{bmatrix},$ verify that
$(\text{A}-\text{B})^\text{T}=\text{A}^\text{T}-\text{B}^\text{T}$
✓

Answer

Given: $\text{A}=\begin{bmatrix}2&-3\\-7&5\end{bmatrix}$
$\text{A}^\text{T}=\begin{bmatrix}2&-7\\-3&5\end{bmatrix}$
$\text{B}=\begin{bmatrix}1&0\\2&-4\end{bmatrix}$
$\text{B}^\text{T}=\begin{bmatrix}1&2\\0&-4\end{bmatrix}$
Given,
$(\text{A}-\text{B})^\text{T}=\text{A}^\text{T}-\text{B}^\text{T}$
$\Rightarrow\begin{pmatrix} \begin{bmatrix} 2&-3\\-7&5\end{bmatrix}-\begin{bmatrix} 1&0\\2&-4\end{bmatrix}\end{pmatrix}^\text{T}$ $=\begin{bmatrix}2&-7\\-3&5\end{bmatrix}-\begin{bmatrix}1&2\\0&-4\end{bmatrix}$
$\Rightarrow\begin{pmatrix}\begin{bmatrix}2-1&-3-0\\-7-2&5+4\end{bmatrix}^\text{T}\end{pmatrix}$ $=\begin{bmatrix}2-1&-7-2\\-3-0&5+4\end{bmatrix}$
$\Rightarrow\begin{pmatrix}\begin{bmatrix}1&-3\\-9&9\end{bmatrix}\end{pmatrix}^\text{T}=\begin{bmatrix}1&-9\\-3&9\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1&-9\\-3&9\end{bmatrix}=\begin{bmatrix}1&-9\\-3&9\end{bmatrix}$
$\therefore\ \text{LHS}=\text{RHS}$

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