Solution:
Solution
When a matrix of order $$ m \times a $$ is multiplied with a matrix of order $$ a \times n $$, the resulting matrix is of order $$ m \times n $$
As product matrix $$ A{B}^{T} $$ will have the order $$ 2 \times 3 $$, we get $$ m = 2 ; n = 3 $$
As A has only one column, $$ => a = 1 $$
Thus $$ {B}^{T} $$ has the order $$ a \times n $$ or $$ 1 \times 3 $$,
As $$ {B}^{T} $$ is of order $$ 1 \times 3 $$, then $$ {B} $$ will have the order $$ 3 \times 1 $$
So, B will also have one column
⇒ 1 - cω - aω + acω ≠ 0
⇒ (1 - cω) (1 - aω) ≠ 0
$$a\ \ne \ \cfrac{1}{\omega },\ c\ \ne \ \cfrac{1}{\omega }$$
⇒ a = ω, c = ω and b⊂ {ω, ω2}
Therefore, number of distinct solutions = 2