Question

A matrix X has a + b rows and a + 2 columns while the matrix Y has b + 1 rows and a + 3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal.

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ \left[\begin{array}{c}X\end{array}\right]{}_{\left(a+b\right)\times \left(a+2\right)} \\ {\left[\begin{array}{c}Y\end{array}\right]}_{\left(b+1\right)\times \left(a+3\right)} \\ \mathrm{Since}XY\mathrm{exists},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{columns}\mathrm{in}X\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{rows}\mathrm{in}Y. \\ \Rightarrow a+2=b+1...\left(1\right) \\ \mathrm{Similarly},\sin ceYX\hspace{0.17em}\mathrm{exists},\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{columns}\mathrm{in}Y\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{rows}\mathrm{in}X. \\ \Rightarrow a+b=a+3 \\ \Rightarrow b=3 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}b\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ a+2=3+1 \\ \Rightarrow a=2 \end{gathered}$$

Since the order of the matrices XY and YX is not same, XY and YX are not of the same type and they are unequal.