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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices2 Marks
Question
Find x, y, a and b if $\begin{bmatrix}2\text{x}-3\text{y}&\text{a}-\text{b}&3\\1&\text{x}+4\text{y}&3\text{a}+4\text{b}\end{bmatrix}=\begin{bmatrix}1&-2&3\\1&6&29\end{bmatrix}$
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Answer

Since the corresponding elements of two equal matrices are equal,
$\begin{bmatrix}2\text{x}-3\text{y}&\text{a}-\text{b}&3\\1&\text{x}+4\text{y}&3\text{a}+4\text{b}\end{bmatrix}=\begin{bmatrix}1&-2&3\\1&6&29\end{bmatrix}$
⇒ 2x - 3y = 1 ...(1)
⇒ x + 4y = 6
⇒ x = 6 - 4y ...(2)
Putting the value of x in eq. (1), we get
2(6 - 4y) - 3y = 1
⇒ 12 - 8y - 3y = 1
⇒ 12 + 11y = 1
⇒ -11y = -11
$\Rightarrow\text{y}=\frac{-11}{-11}=1$
Putting the value of y in eq. (2), we get
x = 6 - 4(1)
⇒ x = 6 - 4
⇒ x = 2
Now,
a - b = -2
⇒ a = -2 + b ...(3)
3a + 4b = 29 ...(4)
Putting the value of a in eq. (4), we get
3(-2 + b) + 4b = 29
⇒ -6 + 3b + 4b = 29
⇒ -6 + 7b = 29
⇒ 7b = 29 + 6
⇒ 7b = 35
$\Rightarrow\text{b}=\frac{35}{7}=5$
Putting the value of b in eq. (3), we get
a = -2 + 5
⇒ a = 3
$\therefore$ a = 3, b = 5, x = 2 and y = 1

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