Question

If A and B are two matrices, match the following to their equivalent expressions.

• A. $A^T+B^T$
• B. $B^T{A}^T$
• C. $A$

Answer 1

Let's take A as A = $\left[\begin{array}{cc}0& 2\\ 3& 1\end{array}\right]$ and B as

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

So, we will have transposes of A and B as:

$A^T$= $\left[\begin{array}{cc}0& 3\\ 2& 1\end{array}\right]$

$B^T$= $\left[\begin{array}{cc}0& 1\\ 1& 2\end{array}\right]$


Three of the properties of transpose are as below:
(T stands for "transpose " )

1.$(A^T{)}^T$ = $A$
Verification:
We have $A^T$= $\left[\begin{array}{cc}0& 3\\ 2& 1\end{array}\right]$ .

So, $(A^T{)}^T$ = $\left[\begin{array}{cc}0& 2\\ 3& 1\end{array}\right]$ = A

Thus, the property is true.

2.$(A+B{)}^T$ = $A^T+B^T$
Verification:

$A+B=\left[\begin{array}{cc}0& 3\\ 4& 3\end{array}\right]$

$(A+B{)}^T=\left[\begin{array}{cc}0& 4\\ 3& 3\end{array}\right]$

$A^T+B^T=\left[\begin{array}{cc}0& 4\\ 3& 3\end{array}\right]$

We see that $(A+B{)}^T$ = $A^T+B^T$.

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

Verification:

$AB=\left[\begin{array}{cc}2& 4\\ 1& 5\end{array}\right]$

$(AB{)}^T=\left[\begin{array}{cc}2& 1\\ 4& 5\end{array}\right]$

$B^T{A}^T=\left[\begin{array}{cc}2& 1\\ 4& 5\end{array}\right]$

We see that $(AB{)}^T$ = $B^T{A}^T$