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Algebra of Matrices — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices2 Marks
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 $Y X$ exist. Find $a$ and $b$. Can you say $X Y$ and $Y X$ are of the same type? Are they equal.
✓

Answer

Here,
${[X]_{(a+b)} \times(a+2)}$
${\left[Y_{(b+1)} \times(a+3)\right.}$
Since XY exists, the number of columns in X is equal to the number of rows in Y .
$\Rightarrow a+2=b+1$
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
$a+2 \approx 3+1$
$\Rightarrow a=2$
Since the order of the matrices $X Y$ and $Y X$ is not same, $X Y$ and $Y X$ are not of the same type and they are unequal.

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