Question

From the following, find the correct relation

• A. (AB)' = A'B'
• B. (AB)' = B'A'
• C. AB = A'B'
• D. (AB)' = BA

Answer 1

The correct option is B

(AB)' = B'A'

Consider an example. Let A = $\left[\begin{array}{cc}1& 2\\ 1& 0\end{array}\right]$ and B = $\left[\begin{array}{cc}2& 1\\ 0& 0\end{array}\right]$

AB = $\left[\begin{array}{cc}2& 1\\ 2& 1\end{array}\right]$ ……………………….. (1)

(AB)’ = $\left[\begin{array}{cc}2& 2\\ 1& 1\end{array}\right]$…………………………… (2)

BA = $\left[\begin{array}{cc}3& 4\\ 0& 0\end{array}\right]$ ………………………… (3)

A’ = $\left[\begin{array}{cc}1& 1\\ 2& 0\end{array}\right]$and B’ = $\left[\begin{array}{cc}2& 0\\ 1& 0\end{array}\right]$

A’B’ = $\left[\begin{array}{cc}3& 0\\ 4& 0\end{array}\right]$ ………………………….. (4)

B’A’ = $\left[\begin{array}{cc}2& 2\\ 1& 1\end{array}\right]$ ………………………. (5)

From (1), (2), (3), (4) and (5)

(AB)’ $\ne$ A’B’

(AB)’ = B’A’

AB $\ne$ A’B’

(AB)’ $\ne$ BA