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Home  >>  CBSE XII  >>  Math  >>  Matrices
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If $A=\begin{bmatrix}2 & 1\end{bmatrix}\;B=\begin{bmatrix}5 & 3 & 4\\8 & 7 & 6\end{bmatrix}\;andC=\begin{bmatrix}-1 & 2 & 1\\1 & 0 & 2\end{bmatrix},verify\;that\;A(B+C)=(AB+AC)$

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Toolbox:
  • If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
  • The sum / difference $A(+/-)B$ of two $m$-by-$n$ matrices $A$ and $B$ is calculated entrywise: $(A (+/-) B)_{i,j} = A_{i,j} +/- B_{i,j}$ where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
Step1:
Given
$A=\begin{bmatrix}2 & 1\end{bmatrix}$
$B=\begin{bmatrix}5 & 3& 4\\8 & 7 & 6\end{bmatrix}$
$C=\begin{bmatrix}-1 & 2&1\\1 & 0& 2\end{bmatrix}$
LHS:-
$A(B+C)=\begin{bmatrix}2 & 1\end{bmatrix}\begin{bmatrix}5 & 3& 4\\8 & 7 & 6\end{bmatrix}\begin{bmatrix}-1 & 2&1\\1 & 0& 2\end{bmatrix}$
$\;\;\;\;\qquad=\begin{bmatrix}2 & 1\end{bmatrix}\begin{bmatrix}5-1 & 3+2& 4+1\\8+1 & 7+0 & 6+2\end{bmatrix}$
$\;\;\;\;\qquad=\begin{bmatrix}2 & 1\end{bmatrix}\begin{bmatrix}4 & 5&5\\9 & 7 & 8\end{bmatrix}$
$\;\;\;\;\qquad=\begin{bmatrix}2(4)+1(9) & 2(5)+1(7)&2(5)+1(8)\end{bmatrix}$
$\;\;\;\;\qquad=\begin{bmatrix}8+9& 10+7&10+8\end{bmatrix}$
$\;\;\;\;\qquad=\begin{bmatrix}17& 17&18\end{bmatrix}$
Step2:
RHS:-
AB+AC
AB=$\begin{bmatrix}2& 1\end{bmatrix}\begin{bmatrix}5& 3& 4\\8 & 7 & 6\end{bmatrix}$
$\Rightarrow \begin{bmatrix}2(5)+1(8)& 2(3)+1(7) & 2(4)+1(6)\end{bmatrix}$
$\Rightarrow \begin{bmatrix}10+8& 6+7 & 8+6\end{bmatrix}$
$\Rightarrow \begin{bmatrix}18& 13 & 14\end{bmatrix}$
$AC=\begin{bmatrix}2 & 1\end{bmatrix}\begin{bmatrix}-1 & 2 & 1\\1 & 0 & 2\end{bmatrix}$
$\Rightarrow \begin{bmatrix}2(-1)+1(1)& 2(2)+1(0) & 2(1)+1(2)\end{bmatrix}$
$\Rightarrow \begin{bmatrix}-2+1& 4+0 & 2+2\end{bmatrix}$
$\Rightarrow \begin{bmatrix}-1& 4 & 4\end{bmatrix}$
AB+AC=$\begin{bmatrix}18& 13 & 14\end{bmatrix}+ \begin{bmatrix}-1& 4 & 4\end{bmatrix}$
$\;\;\;\;\qquad=\begin{bmatrix}17& 17& 18\end{bmatrix}$
$\Rightarrow LHS=RHS.$
$\Rightarrow A(B+C)=AB+AC.$
answered Mar 23, 2013 by sharmaaparna1
 

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