If $A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\end{array}\right]$ and $B=\left[\begin{array}{lll}b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23}\end{array}\right]$,
then which one of the following is True?
$\mathrm{A}^{\mathrm{T}} \mathrm{BB}^{\mathrm{T}} \mathrm{A}=\mathrm{B}^{\mathrm{T}} A \mathrm{~A}^{\mathrm{T}} \mathrm{B}$
The orders of $A^T B^T A$ and $B^T A A^T B$ are equal
The orders of $\mathrm{A}+\mathrm{B}, \mathrm{A}^T \mathrm{~B}, \mathrm{BA}^{\mathrm{T}}$ are equal
Rank of A and B are equal
Solution
Give matrices are
$A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\end{array}\right] ; \quad B=\left[\begin{array}{lll}b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23}\end{array}\right]$
Here matrix $A$ and $B$ have order $2 \times 3$.
$\therefore$ Order of $A^T$ and $B^T$ is $3 \times 2$.
Now, order of $A_{3 \times 2}^T B_{2 \times 3} B_{3 \times 2}^T A_{2 \times 3}=3 \times 3$ ...(i)
And order of $B_{3 \times 2}^T A_{2 \times 3} A_{3 \times 2}^T B_{2 \times 3}=3 \times 3$ ...(ii)
So, from eqn. (i) and (ii)
Order of $A^T B B^T A=$ Order of $B^T A A^T B$