Please Solve RD Sharma Class 12 Chapter Algebra of Matrices Exercise 4.1 Question 19 Maths Textbook Solution.

Answer: $A$ and $B$ are not equal for any value of $y$

Given: $A=\begin{bmatrix} 2x+1 &2y \\ 0 & y^{2}+2 \end{bmatrix}$  and

$B=\begin{bmatrix} x+3 &y^{2}+2 \\ 0 & -6 \end{bmatrix}$

We have to find out the value of $x$  and $y$
Hint: We will use equality of matrices.
Solution:   Here $A=B$
Since equal matrix have all corresponding entries equal. So,

$2x+1=x+3$                                                                                                                                           ….. (i)

$2y=y^{2}+2$                                                                                                                                             ….. (ii)

$y2-5y=-6$                                                                                                                                           ….. (iii)

Solving equation (i), We get

$2x+1=x+3\\\Rightarrow 2x-x=3-1\\\Rightarrow x=2$

Solving equation (ii), We get

$2y=y^{2}+2\\\Rightarrow y^{2}-2y+2=0\\\Rightarrow D=b^{2}-4ac$

$\! \! \! \! \! \! \! \! \! =\left ( -2 \right )^{2}-4\left ( 1 \right )\left ( 2 \right )\\=4-8\\=-4$

Here $D< 0$
So, there is no real value of $y$  from equation (ii)
Solving equation (iii), We get

$y^{2}5y=-6\\\Rightarrow y^{2}-5y+6=0\\\left ( y-3 \right )\left ( y-2 \right )=0\\y=3\: or\: 2$

From solution of equation (i) (ii) and (iii)
we can say that $A$ and $B$ cannot equal for any value of  $y$