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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices5 Marks
Question
Find the matrix A such that
$\begin{bmatrix}1&1\\0&1\end{bmatrix}\text{A}=\begin{bmatrix}3&3&5\\1&0&1\end{bmatrix}$
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Answer

Let $\text{A}=\begin{bmatrix}\text{x}&\text{y}&\text{z}\\\text{a}&\text{b}&\text{c}\end{bmatrix}$
$\Rightarrow\begin{bmatrix}1&1\\0&1\end{bmatrix}\begin{bmatrix}\text{x}&\text{y}&\text{z}\\\text{a}&\text{b}&\text{c}\end{bmatrix}=\begin{bmatrix}3&3&5\\1&0&1\end{bmatrix}$
$ \Rightarrow\begin{bmatrix}\text{x}+\text{a}&\text{y}+\text{b}&\text{z}+\text{c}\\0+\text{a}&0+\text{b}&0+\text{c}\end{bmatrix}=\begin{bmatrix}3&3&5\\1&0&1\end{bmatrix}$
$ \Rightarrow\begin{bmatrix}\text{x}+\text{a}&\text{y}+\text{b}&\text{z}+\text{c}\\\text{a}&\text{b}&\text{c}\end{bmatrix}=\begin{bmatrix}3&3&5\\1&0&1\end{bmatrix}$
The corresponding elements of two equal matrices are equal.
⇒ x + a = 3 ...(1)
y + b = 3 ...(2)
z + c = 5 ...(3)
⇒ a = 1, b = 0 and c = 1
Putting the value of a in eq. (1), we get
x + 1 = 3
⇒ x = 3 - 1
$\therefore$ x = 2
Putting the value of b in eq. (2), we get
y + b = 3
⇒ y + 0 = 3
$\therefore$ y = 3
Putting the value of c in eq. (3), we get
z + 1 = 5
⇒ z = 5 - 1
$\therefore$ z = 4
$\therefore\ \text{A}=\begin{bmatrix}2&3&4\\1&0&1\end{bmatrix}$

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