QS-017 · Linear Algebra · Easy
Non-Commutative Matrix Expansion
Suppose $A$ and $B$ are two square matrices of the same size. Which of the following statements is NOT necessarily true?
Statement 1: $(A+B)^2=A^2+2AB+B^2$
Statement 2: $(A+B)^2=A^2+AB+BA+B^2$
Statement 3: $(A+B)^2=A(A+B)+B(A+B)$
1. Statement 1
2. Statement 2
3. Statement 3
4. All 3 statements
Suppose $A$ and $B$ are two square matrices of the same size. Which of the following statements is NOT necessarily true?
Statement 1: $(A+B)^2=A^2+2AB+B^2$
Statement 2: $(A+B)^2=A^2+AB+BA+B^2$
Statement 3: $(A+B)^2=A(A+B)+B(A+B)$
1. Statement 1
2. Statement 2
3. Statement 3
4. All 3 statements
Solution
For matrices, multiplication is distributive, but in general,
$$ AB\neq BA. $$
Expand $(A+B)^2$:
$$ (A+B)^2=(A+B)(A+B) = A(A+B)+B(A+B) $$
Using distributivity,
$$ A(A+B)=A^2+AB $$
and
$$ B(A+B)=BA+B^2. $$
Therefore,
$$ (A+B)^2=A^2+AB+BA+B^2. $$
Now check the statements.
Statement 2:
$$ (A+B)^2=A^2+AB+BA+B^2 $$
This matches the correct expansion, so it is always true.
Statement 3:
$$ (A+B)^2=A(A+B)+B(A+B) $$
This follows directly from distributivity, so it is always true.
Statement 1:
$$ (A+B)^2=A^2+2AB+B^2 $$
This would require
$$ AB+BA=2AB, $$
which is equivalent to
$$ AB=BA. $$
Since matrix multiplication is not necessarily commutative, this need not hold.
Therefore, Statement 1 is NOT necessarily true.
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