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Algebra of Matrices — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSAlgebra of Matrices2 Marks
Question
Let $A$ and $B$ be square matrices of the same order. Does $(A + B)^2 = A^2 + 2AB + B^2$ hold $?$ If not, why $?$
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Answer

$LHS = (A + B)^2$
$= (A + B)(A + B)$
$= A(A + B) + B(A + B)$
$= A^2 + AB + BA + B^2$
We know that a matrix does not have commutative property. So,
$AB \neq BA$
Thus,
$(A + B)^2 \neq A^2 + 2AB + B^2$

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