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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 $\begin{bmatrix}3\text{x}+4\text{y}&2&\text{x}-2\text{y}\\\text{a}+\text{b}&2\text{a}-\text{b}&^-1\end{bmatrix}=\begin{bmatrix}2&2&4\\5&-5&-1\end{bmatrix}$
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Answer

Since the corresponding elements of two equal matrices are equal,
$\begin{bmatrix}3\text{x}+4\text{y}&2&\text{x}-2\text{y}\\\text{a}+\text{b}&2\text{a}-\text{b}&^-1\end{bmatrix}=\begin{bmatrix}2&2&4\\5&-5&-1\end{bmatrix}$
⇒ 3x + 4y = 2 ...(1)
⇒ x - 2y = 4
⇒ x = 4 + 2y ...(2)
Putting the value of x in eq. (1), we get
3(4 + 2y) + 4y = 2
⇒ 12 + 6y + 4y = 2
⇒ 12 + 10y = 2
⇒ 10y = 2 - 12
⇒ 10y = -10
$\Rightarrow\text{y}=\frac{-10}{10}=-1$
Putting the value of y in eq. (2), we get
x = 4 + 2(-1)
⇒ x = 4 - 2 = 2
a + b = 5
⇒ a = 5 - b ...(3)
⇒ 2a - b = -5 ...(4)
Putting the value of a in eq. (4), we get
2(5 - b) - b = -5
⇒ 10 - 2b - b = -5
⇒ 10 - 3b = -5
⇒ -3b = -15
$\Rightarrow\text{b}=\frac{-15}{-3}$
⇒ b = 5
Putting the value of b in eq. (3), we get
a = 5 - 5
⇒ a = 0
$\therefore$ x = 2, y = -1, a = 0 and b = 5

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