Question

From the following find the correct relation [MP PET 1990]
E. $(AB{)}'={A}'{B}$
F. $(AB{)}'={B}'{A}'$
G. ${{A}^{-1}}=\dfrac{adj\,A}{A}$
H. ${{(AB)}^{-1}}={{A}^{-1}}{{B}^{-1}}$

Answer 1

Hint: In this question, we have to check each option satisfies the properties of Matrix. The matrix that results from swapping a matrix's rows and columns is known as the matrix's transpose.

Formula Used: Transpose Property:
$(AB{)}'={B}'{A}'$

Complete step by step solution: Let’s check for option A. We have (AB{)}'={A}'{B}
It’s incorrect.

In option B we have $(AB{)}'={B}'{A}'$
The transpose of the products of two matrices is simply the product of their transposes in the reversed order.
Therefore,
$(AB{)}'={B}'{A}'$
It’s correct.

In option C
We have ${{A}^{-1}}=\dfrac{adj\,A}{A}$
It’s Incorrect. In the inverse formula we use determinant of matrix not the matrix.

For option D we have given ${{(AB)}^{-1}}={{A}^{-1}}{{B}^{-1}}$
Its Incorrect. If A and B are nonsingular matrices, then AB is non-singular and is given by ${{(AB)}^{-1}}={{B}^{-1}}{{A}^{-1}}$.

Option ‘B’ is correct

Note: Keep in mind that multiplying the individual transposes of each matrix in reverse order produces the same result as multiplying the two matrices in transpose.

Additional Information:
Properties of Transpose of a matrix

• 1. The matrix itself is the transpose of the transpose of the matrix: $(A^T)^{T} = A$.

2. The constant multiplied by the transpose of the matrix is equal to the transpose of the matrix times a scalar $(kA)^{T}= kA$.

3. $(A + B)^{T}= A^T + B^T$

4. The transpose of a pair of matrices equals the product of the products of their transposes in the opposite direction: $(AB)^{T} = B^{T} A^{T}$. The product of multiple matrices has the same formula: $(ABC)^{T} = C^{T}B^{T}A^{T}$