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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.
Solution
\[Here, \]
\[\begin{bmatrix}X\end{bmatrix} {}_\left( a + b \right) \times \left( a + 2 \right) \]
\[ \begin{bmatrix}Y\end{bmatrix}_\left( b + 1 \right) \times \left( a + 3 \right) \]
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.
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