If A and B are 2× 2 matrices, then which of the following is true?

Question

If A and B are 2× 2 matrices, then which of the following is true? (A) (A + B)2 = A2 + B2 + 2AB (B)  (A - B)2 = A2 + B2 - 2AB  (C) (A - B)(A + B) = A2 + AB-BA-B2 (D) (A + B)(A - B) = A2 - B2

Tardigrade Admin · Accepted Answer

Given that, A and B are 2×2 matrices. ∴ (A-B)×(A+B) = A× A +A× B -B× A - B× B = A2 - B2 + AB -AB

Q. If $A$ and $B$ are $2 \times 2$ matrices, then which of the following is true?

$(A + B)^{2} = A^{2} + B^{2} + 2 A B$

$(A - B)^{2} = A^{2} + B^{2} - 2 A B$

$(A - B) (A + B) = A^{2} + A B - B A - B^{2}$

$(A + B) (A - B) = A^{2} - B^{2}$

Given that, $A$ and $B$ are $2 \times 2$ matrices. $∴ (A - B) \times (A + B) = A \times A + A \times B - B \times A - B \times B$
$= A^{2} - B^{2} + A B - A B$

Q. If A and B are 2×2 matrices, then which of the following is true?

(A+B)2=A2+B2+2AB

(A−B)2=A2+B2−2AB

(A−B)(A+B)=A2+AB−BA−B2

(A+B)(A−B)=A2−B2