1.

Find x, y, a and b if \(\begin{bmatrix}3x + 4y & 2 & x - 2y \\[0.3em]a + b & 2a - b & -1 \\[0.3em]\end{bmatrix}\) = \(\begin{bmatrix}2& 2 &4 \\[0.3em]5 & -5 & -1 \\[0.3em]\end{bmatrix}\)

Answer»

Given as

\( \begin{bmatrix} 3x + 4y & 2 & x - 2y \\[0.3em] a + b & 2a - b & -1 \\[0.3em] \end{bmatrix}\) = \( \begin{bmatrix} 2& 2 &4 \\[0.3em] 5 & -5 & -1 \\[0.3em] \end{bmatrix}\)

Given as two matrices are equal.

As we know that if two matrices are equal then the elements of each matrices are also equal.

So, by equating them 

3x + 4y = 2 …… (1)

x – 2y = 4 …… (2)

a + b = 5 …… (3)

2a – b = – 5 …… (4)

On multiplying equation (2) by 2 and adding to equation (1),

3x + 4y + 2x – 4y = 2 + 8

⇒ 5x = 10

⇒ x = 2

Substitute this value of x in equation (1)

3 × 2 + 4y = 2

⇒ 6 + 4y = 2

⇒ 4y = 2 – 6

⇒ 4y = – 4

⇒ y = – 1

On adding equation (3) and (4)

a + b + 2a – b = 5 + (– 5)

⇒ 3a = 5 – 5 = 0

⇒ a = 0

Again substitute this value of a in equation (3), 

0 + b = 5

⇒ b = 5

So, a = 0, b = 5, x = 2 and y = – 1


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