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 two matrices are equal,

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

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

∴ 3x + 4y = 2 …… (1) and 

x – 2y = 4 …… (2) and 

a + b = 5 ……(3) 

2a – b = – 5 ……(4) 

Multiplying equation (2) by 2 and adding to equation (1).

3x + 4y + 2x – 4y = 2 + 8

⇒ 5x = 10 

⇒ x = 2

Now, Putting the value of x in equation (1)

3 × 2 + 4y = 2 

⇒ 6 + 4y = 2 

⇒ 4y = 2 – 6 

⇒ 4y = – 4 

⇒ y = – 1

Adding equation (3) and (4),

a + b + 2a – b = 5 + (–5) 

⇒ 3a = 5 – 5 = 0 

⇒ a = 0

Now, Putting the value of a in equation (3) 

0 + b = 5 

⇒ b = 5 

∴ a = 0, b = 5, 

x = 2 and y = – 1


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