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  • Please Solve RD Sharma Class 12 Chapter Algebra of Matrices Exercise 4.2 Question 6 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter Algebra of Matrices Exercise 4.2 Question 6 Maths Textbook Solution.

Answers (1)

Hint:  In LHS, solve (A + B) by adding A and B matrix. Then, add C matrix. and In RHS, first solve (B+C) then add A to it.

Given: To prove:(A + B) + C = A + (B + C)

A = \begin{bmatrix} 2 & 1 & 1\\ 3 & -1& 0\\ 0 & 2 & 4 \end{bmatrix} , B = \begin{bmatrix} 9& 7&-1 \\ 3 & 5 & 4\\ 2 & 1 &6 \end{bmatrix} , C = \begin{bmatrix} 2 & -4 & 3\\ 1 & -1 &0 \\ 9 &4 & 5 \end{bmatrix}
Solution:
LHS = (A + B) + C

           = \left ( \begin{bmatrix} 2 & 1 & 1\\ 3 & -1& 0\\ 0 & 2 & 4 \end{bmatrix} + \begin{bmatrix} 9& 7&-1 \\ 3 & 5 & 4\\ 2 & 1 &6 \end{bmatrix} \right )+ \begin{bmatrix} 2 & -4 & 3\\ 1 & -1 &0 \\ 9 &4 & 5 \end{bmatrix} 

           = \left ( \begin{bmatrix} 2+9 & 1+7 & 1-1\\ 3+3 & -1+5& 0+4\\ 0+2 & 2+1 & 4+6 \end{bmatrix} \right )+ \begin{bmatrix} 2 & -4 & 3\\ 1 & -1 &0 \\ 9 &4 & 5 \end{bmatrix} 

          = \begin{bmatrix} 2+9+2 & 1+7 -4& 1-1+3\\ 3+3+1 & -1+5-1& 0+4+0\\ 0+2+9 & 2+1 +4& 4+6 +5\end{bmatrix} 

          = \begin{bmatrix} 13 & 4& 3\\ 7 & 3& 4\\ 11 & 7& 15\end{bmatrix}                                                                                 … (1)

 

RHS = A + (B + C)

= \begin{bmatrix} 2 & 1 & 1\\ 3 & -1& 0\\ 0 & 2 & 4 \end{bmatrix} \left ( \begin{bmatrix} 9& 7&-1 \\ 3 & 5 & 4\\ 2 & 1 &6 \end{bmatrix} +\begin{bmatrix} 2 & -4 & 3\\ 1 & -1 &0 \\ 9 &4 & 5 \end{bmatrix} \right ) 

= \begin{bmatrix} 2 & 1 & 1\\ 3 & -1& 0\\ 0 & 2 & 4 \end{bmatrix} \left ( \begin{bmatrix} 9+2& 7-4&-1+3 \\ 3+1 & 5-1 & 4+0\\ 2+9 & 1+4 &6 +5\end{bmatrix} \right ) 

= \begin{bmatrix} 2+9+2&1+ 7-4&1-1+3 \\3+ 3+1 &-1+0+ 5-1 & 0+4+0\\0+ 2+9 & 2+1+4 &4+6 +5\end{bmatrix} 

= \begin{bmatrix} 13&4&3 \\7 &3 & 4\\11 & 7 &15\end{bmatrix}                                                                               … (2)
From (1) and (2)

LHS = RHS

Hence,(A + B) + C = A + (B + C) is proved.

Posted by

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