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#### Need solution for RD Sharma maths class 12 chapter Algebra of matrices exercise 4.3 question 17 (i) maths textbook solution

Hence, verify the distribution of matrix multiplication over matrix addition $A(B+C)=A B+ A C$

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:$A=\left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right], B=\left[\begin{array}{cc}-1 & 0 \\ 2 & 1\end{array}\right], C=\left[\begin{array}{cc}0 & 1 \\ 1 & -1\end{array}\right]$

Consider,

$A(B+C) \\\\ =\left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right]\left(\left[\begin{array}{cc}-1 & 0 \\ 2 & 1\end{array}\right]+\left[\begin{array}{ll}0 & 1 \\ 1 & -1\end{array}\right]\right) \\\\\\ = \left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right]\left(\left[\begin{array}{cc}-1+0 & 0+1 \\ 2+1 & 1-1\end{array}\right]\right)=\left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right]\left[\begin{array}{cc}-1 & 1 \\ 3 & 0\end{array}\right] \\\\$

$\begin{bmatrix} -1-3 & 1+0 \\ -0+6& 0+0 \end{bmatrix}$

$A(B+C)=\left[\begin{array}{cc} -4 & 1 \\ 6 & 0 \end{array}\right]$

$A B+A C=\left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right]\left[\begin{array}{cc}-1 & 0 \\ 2 & 1\end{array}\right]+\left[\begin{array}{cc}1 & -1 \\ 0 & 2\end{array}\right]\left[\begin{array}{cc}0 & 1 \\ 1 & -1\end{array}\right]\\\\\\=\left[\begin{array}{cc}1(-1)+(-1)(2) & 1(0)+(-1)(1) \\0(-1)+2(2) & 0(0)+2(1)\end{array}\right]+\left[\begin{array}{cc}(1)(0)+(-1)(1) & 1(1)+(-1)(-1) \\ 0(0)+2(1) & 0(1)+2(-1)\end{array}\right] \\\\\\=\left[\begin{array}{ll}-1-2 & 0-1 \\ -0+4 & 0+2\end{array}\right]+\left[\begin{array}{ll}0-1 & 1+1 \\ 0+2 & 0-2\end{array}\right]$

$A B+A C=\left[\begin{array}{cc} -4 & 1 \\ 6 & 0 \end{array}\right]$

From equation i & equation ii

$A(B+C)=A B+ A C$