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  • Provide solution for rd sharma math class 12 chapter Algebra of matrices exercise 4.3 question 48 sub question (ii)

Provide solution for rd sharma math class 12 chapter Algebra of matrices exercise 4.3 question 48 sub question (ii)

Answers (1)

Answer:

A=\left[\begin{array}{cc}1 & -2 \\ 2 & 0\end{array}\right]

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}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]_{2 \times 3}=\left[\begin{array}{ccc}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right]_{2 \times 3}

The matrix given on the RHS of the equation is a {2 \times 3} matrix and the matrix given on the LHS of the equation is {2 \times 3} So, matrix A has to be {2 \times 2} matrix.

Now, let

A=\left[\begin{array}{ll}a & c \\ b & d\end{array}\right]

we have, 

\\\\ \Rightarrow\left[\begin{array}{ll}a & c \\ b & d\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]=\left[\begin{array}{ccc}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right] \\\\ \Rightarrow \left[\begin{array}{ccc}a+4 c & 2 a+5 c & 3 a+6 c \\ b+4 d & 2 b+5 d & 3 b+6 d\end{array}\right]=\left[\begin{array}{ccc}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right]

Equating the corresponding elements of the two matrices, we have

a+4 c=-7,2 a+5 c=-8,3 a+6 c=-9, b+4 d=2,2 b+5 d=4,3 b+6 d=6

Now,

\\\\ \Rightarrow a+4 c=-7 \quad \ldots (i) \\\\ \Rightarrow a=-7-4 c\\\\ 2 a+5 c=-8 \quad \ldots (ii)

Put the value of a in equation ii

\begin{array}{l} \Rightarrow 2(-7-4 c)+5 c=-8 \\\\ \Rightarrow-14-8 c+5 c=-8 \\\\ \Rightarrow-3 c=6 \\\\ \Rightarrow c=-2 \end{array}

Now, put value of a in a=-7-4c

\\\\ \Rightarrow a=-7-4(-2)=-7+8=1 \\\\ \Rightarrow a=1

Using (iii) in (iv)

\\\\ \Rightarrow 4-8 d+5 d=4\\\\ \Rightarrow-3 d=0\\\\ \Rightarrow d=0

Now use the value of d=0 in b+4d=2                     

\therefore b=2-4(0)=2

Thus, a=1, b=2, c=-2, d=0

Hence, the required matrix A is

\left[\begin{array}{cc}1 & -2 \\ 2 & 0\end{array}\right]

Posted by

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