Question

If A is a $2\times 3$ matrix and B is $3\times 2$ matrix then the order of ${\left(AB\right)}^T$ is equal to the order of _______

• A. $AB$
• B. $A^T{B}^T$
• C. BA
• D. All of these

Answer 1

Answer: (A)

$AB$

NOTE: $P_{(m\times n)}\times Q_{(n\times a)}$ Then order of resultant is $(m\times a)$

Given

$A\to (2\times 3)$

$B\to (3\times 2)$

Order of ${\left(AB\right)}^T$

$={[A_{(2\times 3)}\times B_{(3\times 2)}]}^T$

$={\left[A{B}_{(2\times 2)}\right]}^T$

Order of ${\left(AB\right)}^T$ is $(2\times 2)$

Option A: $AB$

$A_{(2\times 3)}\times B_{(3\times 2)}$

$A{B}_{(2\times 2)}=$ order ${\left(AB\right)}_{(2\times 2)}^T$

$\therefore$Option satisfied

Option B: $A^T{B}^T$

$A_{(3\times 2)}^T{B}_{(2\times 3)}^T$

$A^T{B}_{(3\times 3)}^T\ne$ order of${\left(AB\right)}^T$

Not satisfied