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

\[Here, \]

\[\begin{bmatrix}X\end{bmatrix} {}_\left( a + b \right) \times \left( a + 2 \right) \]

\[ \begin{bmatrix}Y\end{bmatrix}_\left( b + 1 \right) \times \left( a + 3 \right) \]

Since XY exists, the number of columns in X is equal to the number of rows in Y.

\[ \Rightarrow a + 2 = b + 1 . . . \left( 1 \right)\]

\[\]

Similarly,  since YX  exists, the number of columns in Y is equal to the number of rows in X . 

\[ \Rightarrow a + b = a + 3\]

\[ \Rightarrow b = 3\]

Putting the value of b in   (1),  we get

\[a + 2 = 3 + 1\]

\[ \Rightarrow a = 2\]

\[\]

\[\]

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.