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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat 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 $YX$ exist. Find $a$ and $b.$ Can you say $XY$ and $YX$ are of the same type$?$ Are they equal.
✓

Answer

Here,
$[X]_{(a+b) \times (a+2)}$
$[Y]_{(b+1) \times (a+3)}$
Since $XY$ exists, the number of columns in $X$ is equal to the number of rows in $Y.$
$⇒ a + 2 = b + 1 ...(1)$
Similarly, since $YX$ exists, the number of columns in $Y$ is equal to the number of rows in $X.$
$⇒ a + b = a + 3$
$⇒ b = 3$
Putting the value of b in $(1),$ we get
$a + 2 ≈ 3 + 1$
$⇒ a = 2$
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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