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  • Please Solve RD Sharma Class 12 Chapter 6 Adjoint and Inverese of Matrices Exercise 6.1 Question 5 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 6 Adjoint and Inverese of Matrices Exercise 6.1 Question 5 Maths Textbook Solution.

Answers (1)

Answer:

\operatorname{Adj}(A)=3 A^{T}

Hint:

Here, we use basic concept of determinant and adjoint of matrix.

Given:

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

Solution:

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

Let’s find cofactors  C_{ij}=\left ( -1 \right )^{i+j}                                      

 \begin{aligned} &C_{11}=+(1-4)=-3 \\ &C_{12}=-(2+4)=-6 \\ &C_{13}=+(-4-2)=-6 \\ &C_{21}=-(-2-4)=6 \\ &C_{22}=+(-1+4)=3 \\ &C_{23}=-(2+4)=-6 \\ &C_{31}=+(4+2)=6 \\ &C_{32}=-(2+4)=-6 \\ &C_{33}=+(-1+4)=3 \end{aligned}

C_{i j}=\left[\begin{array}{ccc} -3 & -6 & -6 \\ 6 & 3 & -6 \\ 6 & -6 & 3 \end{array}\right]

So,

Adj\left ( A \right ) = Transpose of  C_{ij}  

\operatorname{Adj}(A)=\left[\begin{array}{ccc} -3 & 6 & 6 \\ -6 & 3 & -6 \\ -6 & -6 & 3 \end{array}\right]                    (1)

3 A^{T}=3 \times\left[\begin{array}{ccc} -1 & 2 & 2 \\ -2 & 1 & -2 \\ -2 & -2 & 1 \end{array}\right]=\left[\begin{array}{ccc} -3 & 6 & 6 \\ -6 & 3 & -6 \\ -6 & -6 & 3 \end{array}\right]                 (2)

Here, from equation (1) and (2)

Clearly see that,

3 A^{T}=A d j(A)

Hence, proved

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