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  • Need solution for RD Sharma Maths Class 12 Chapter Algebra of Matrices Excercise 4.4 Question 6 Subquestion (ii).

Need solution for RD Sharma Maths Class 12 Chapter Algebra of Matrices Excercise 4.4 Question 6 Subquestion (ii).

Answers (1)

Answer: \left (AB \right )^{T}=B^{T}A^{T}

Given: A=\begin{bmatrix} 1 &3 \\ 2 & 4 \end{bmatrix},B=\begin{bmatrix} 1 &4 \\ 2& 5 \end{bmatrix}

To prove: \left (AB \right )^{T}=B^{T}A^{T}

Hint: The A^{T} of matrix  A can be obtained by reflecting the elements along it’s main diagonal.

Solution:

               A^{T}=\begin{bmatrix} 1 &2 \\ 3 & 4 \end{bmatrix},B^{T}=\begin{bmatrix} 1 &2\\ 4& 5 \end{bmatrix}

               \left (AB \right )^{T}=B^{T}A^{T}

               \left ( \begin{bmatrix} 1 &3 \\ 2& 4 \end{bmatrix}\begin{bmatrix} 1 &4 \\ 2 & 5 \end{bmatrix} \right )^{T}=\begin{bmatrix} 1 &2 \\ 4& 5 \end{bmatrix}\begin{bmatrix} 1 &2 \\ 3& 4 \end{bmatrix}

               \begin{bmatrix} 1+6 &4+15 \\ 2+8 & 8+20 \end{bmatrix}^{T}=\begin{bmatrix} 1+6 & 2+8\\ 4+15 &8+20 \end{bmatrix}

               \begin{bmatrix} 7 & 19\\ 10 & 28 \end{bmatrix}^{T}=\begin{bmatrix} 7 &10 \\ 19& 28 \end{bmatrix}

               \begin{bmatrix} 7 & 19\\ 10 & 28 \end{bmatrix}=\begin{bmatrix} 7 &19 \\ 19& 28 \end{bmatrix}

               ∴LHS=RHS

Hence, \left (AB \right )^{T}=B^{T}A^{T} is proved.

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