Question

$(AB{)}^T=A^T{B}^T$. The statement is

• A. True
• B. False

Answer 1

The correct option is B

False

We have discussed already that $(AB{)}^T=B^T{A}^{T}not{A}^T{B}^T$.

Now to prove this we will show that for any row i and column j the (i, j) entry on L.H.S = (i,j) entry on the R.H.S.

The (i,j) entry on L.H.S is $(AB{)}_{i,j}^T$ which is the same as (j,i) entry of AB as $(AB{)}^T$ is nothing but transpose of AB.

So $(AB{)}_{i,j}^T=(AB{)}_{j,i}$

The j, i entry of AB is (row j of A). (Column i of B)(Refer to multiplication of matrices).

On the other hand the i, j entry of R.H.S is the i, j entry of the product $B^T{A}^T$.

This can be expressed as (row i of $B^T$) . (Column j of $A^T$).

Which is same as saying (column I of B). (row j of A).

Which is same as LHS. So we have proved that i, j entries on two sides are the dot product of same things. So $(AB{)}^T=B^T{A}^T$.