#### Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 64 math

Answer: a=5, b=4 and order of XY and YX are not the same and they are not equal but both are square matrices

Hint: Matrix multiplication is only possible, when the number of columns of first matrix is equal to the number of rows of second matrix.

Given: Matrix X has a+b rows and a+2 columns. Matrix y has b+1 rows and a+3 column both the matrices XY and  YX exist.

So, order of matrix $X=(a+b)(a+2)$ order of matrix $Y=(b+1)(a+3)$

Multiplication of matrix YX exists, when the number of columns of Y is equal to the number of rows of X.

$\begin{array}{l} \text { So, } Y_{(b+1) \times(a+3)}{X}_{(a+b) \times(a+2)} \text { exists }\\\\ a+3=a+b\\\\ b=3 \quad \ldots (i) \end{array}$

Multiplication of matrix XY exists, when the number of columns of X is equal to the number of rows of Y.

$\begin{array}{l} X_{(a+b) \times(a+2)} Y_{(b+1) \times(a+3)} \\\\ (a+2)=(b+1) \\\\ a-b=-1 \\\\ a-3=-1 \quad \text { [ using (i) ] }\\\\ a=2 ... (ii)\\ \end{array}$

So order of  $X=(a+b) \times(a+2)$

\\ \qquad \begin{aligned} &=(2+3) \times(2+2) \\ &=5 \times 4 \\ &=(3+1) \times(2+3) \\ &=4 \times 5 \end{aligned}

Order of   $Y=(b+1) \times(a+3)$

$\\ \begin{array}{l} \\=(3+1) \times(2+3) \\ =4 \times 5 \\ \end{array}$

Order of   $X_{5 \times 4} Y_{4 \times 5}=5 \times 5 \\$

Order of    $Y_{4 \times 5} X_{5 \times 4}=4 \times 4$

So, order of XY and YX are not same and they are not equal but both XY and YX are square matrices.