Question

Which of the following statements is true regarding the transpose of a matrix?

• A. The transpose of a matrix will have same order as the matrix itself.
• B. For any 2 matrices $A$ and $B$, if $AB$ exists that means $B^T{A}^T$ also exists.
• C. For any 2 matrices $A$ and $B$, if $AB$ exists that means $A^T{B}^T$ also exists.
• D. The multiplication of a matrix and it's transpose is not possible.

Answer 1

The correct option is B For any 2 matrices $A$ and $B$, if $AB$ exists that means $B^T{A}^T$ also exists.

Let there be matrices A and B with order a x b and c x d respectively.
AB exists. That means b = c
Order of $A^T$ and $B^T$ is b x a and d x c respectively.
As b = c, $B^T{A}^T$ exists.
For any 2 matrices $A$ and $B$, if $AB$ exists that means $B^T{A}^T$ also exists.