For what value of x and y are the following matrices equal?\(A=\begin{

1.	For what value of x and y are the following matrices equal?\(A=\begin{bmatrix}2x+1& 2y \\[0.3em]0 & y^2-2y \\[0.3em]\end{bmatrix},\)\(B=\begin{bmatrix}x+3& y^2+2 \\[0.3em]0 & -6 \\[0.3em]\end{bmatrix}\)
Answer» Given two matrices are equal. i.e, A = B. \(\begin{bmatrix}2x+1& 2y \\[0.3em]0 & y^2-2y \\[0.3em]\end{bmatrix}\)\(=\begin{bmatrix}x+3& y^2+2 \\[0.3em]0 & -6 \\[0.3em]\end{bmatrix}\) We know that if two matrices are equal, then the elements of each matrices are also equal. ∴2x + 1 = x + 3 ⇒2x – x = 3 – 1 ⇒x = 2 …(1) And 2y = y² + 2 ⇒ y² – 2y + 2 = 0 ⇒ y = \(\frac{-2±\sqrt{4-8}}{2}\) ⇒ y = \(\frac{-2±2i}{2}\) ⇒ y = \(\frac{2(-1±i)}{2}\) ⇒ y = -1 ± i (No real solutions) … (2) And y² – 5y = – 6 ⇒ y² – 5y + 6 = 0 ⇒ y² – 3y – 2y + 6 = 0 ⇒ y(y – 3) – 2(y – 3) = 0 ⇒ (y – 3)(y – 2) = 0 ⇒ y = 3 or 2 … (3) ∴From the above equations we can say that A and B can’t be equal for any value of y.

For what value of x and y are the following matrices equal?\(A=\begin{bmatrix}2x+1& 2y \\[0.3em]0 & y^2-2y \\[0.3em]\end{bmatrix},\)\(B=\begin{bmatrix}x+3& y^2+2 \\[0.3em]0 & -6 \\[0.3em]\end{bmatrix}\)

Answer»

Given two matrices are equal.

i.e, A = B.

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

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

∴2x + 1 = x + 3

⇒2x – x = 3 – 1

⇒x = 2 …(1)

And 2y = y² + 2

⇒ y² – 2y + 2 = 0

⇒ y = \(\frac{-2±\sqrt{4-8}}{2}\)

⇒ y = \(\frac{-2±2i}{2}\)

⇒ y = \(\frac{2(-1±i)}{2}\)

⇒ y = -1 ± i

(No real solutions) … (2)

And y² – 5y = – 6

⇒ y² – 5y + 6 = 0

⇒ y² – 3y – 2y + 6 = 0

⇒ y(y – 3) – 2(y – 3) = 0

⇒ (y – 3)(y – 2) = 0

⇒ y = 3

or 2 … (3)

∴From the above equations we can say that A and B can’t be equal for any value of y.