Consider the following information regarding the number of men and women workers in three factories $I,\,II$ and $III$

Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?

If $A = \left[ {\begin{array}{*{20}{c}}{ab}&{{b^2}}\\{ – {a^2}}&{ – ab}\end{array}} \right]$ and ${A^n} = O$, then the minimum value of $n$ is

If the matrix $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1\end{array}\right]$ satisfies the equation $A ^{20}+\alpha A ^{19}+\beta A =\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1\end{array}\right]$ for some real numbers $\alpha$ and $\beta$, then $\beta-\alpha$ is equal to …….. .

$\left[ {\begin{array}{*{20}{c}}7&1&2\\9&2&1\end{array}} \right]\,\left[ \begin{array}{l}3\\4\\5\end{array} \right] + 2\left[ \begin{array}{l}4\\2\end{array} \right]$ is equal to

$AB = 0$, if and only if