Find the value of x and y, if 2 [(1,3)(0,x)]+ [(y,0)(1,2)]= [(5,6)(1,8)]. ​​​

Question

Find the value of x and y, if 2\(\begin{bmatrix} 1& 3 \\[0.3em] 0 & x \\[0.3em] \end{bmatrix}\)+\(\begin{bmatrix} y& 0 \\[0.3em] 1 & 2 \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix} 5& 6 \\[0.3em] 1 & 8 \\[0.3em] \end{bmatrix}.\)

Answer 1

We are given that,

2\(\begin{bmatrix} 1& 3 \\[0.3em] 0 & x \\[0.3em] \end{bmatrix}\)+\(\begin{bmatrix} y& 0 \\[0.3em] 1 & 2 \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix} 5& 6 \\[0.3em] 1 & 8 \\[0.3em] \end{bmatrix}\)

We need to find the value of x and y.

Taking Left Hand Side (LHS) matrix of the equation,

LHS,

 2\(\begin{bmatrix} 1& 3 \\[0.3em] 0 & x \\[0.3em] \end{bmatrix}\)+\(\begin{bmatrix} y& 0 \\[0.3em] 1 & 2 \\[0.3em] \end{bmatrix}\)

Multiplying the scalar, 2 by each element of the matrix \(\begin{bmatrix} 1& 3 \\[0.3em] 0 & x \\[0.3em] \end{bmatrix}\),

Adding the corresponding elements,

Equate LHS to Right Hand Side (RHS) equation,

\(\begin{bmatrix} 2+y& 6 \\[0.3em] 1 & 2x+2 \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix} 5& 6 \\[0.3em] 1 &8 \\[0.3em] \end{bmatrix}\)

We know that if we have,

\(\begin{bmatrix} a_{11}& a_{12} \\[0.3em] a_{21} & a_{22} \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix} b_{11}& b_{12} \\[0.3em] b_{21} & b_{22} \\[0.3em] \end{bmatrix}\)

This implies, 

a₁₁ = b_11, a₁₂ = b₁₂, a₂₁ = b₂1 and a₂₂ = b₂₂

Similarly, 

The corresponding elements of two matrices are equal,

2 + y = 5 …(i) 

6 = 6 

1 = 1 

2x + 2 = 8 …(ii)

We have equations (i) and (ii) to solve for x and y. 

From equation (i),

2 + y = 5 

⇒ y = 5 – 2 

⇒ y = 3

From equation (ii),

2x + 2 = 8 

⇒ 2x = 8 – 2 

⇒ 2x = 6 

⇒ x = 3 

Thus, 

We have x = 3 and y = 3.