If A and B are square matrices of the same order, then \((A+B)(A-B)\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(A^2 - B^2 + BA - AB\)
By the distributive law, keeping the order of every product:
\((A + B)(A - B) = A(A - B) + B(A - B)\)
\(= A^2 - AB + BA - B^2\)
\(= A^2 - B^2 + BA - AB\)
In general \(AB \ne BA\) (matrix multiplication is not commutative), so \(BA - AB\) does not cancel.
Why the other options are incorrect
Option (1) \(A^2 - B^2\) — Incorrect. This holds only when \(AB = BA\), which is not given.
Option (3) \(A^2 - BA - AB - B^2\) — Incorrect. The term BA comes from \(B \cdot A\) with a plus sign, not a minus sign.
Option (4) \(B^2 + A^2 + AB - BA\) — Incorrect. The expansion has \(-B^2\), \(-AB\) and \(+BA\); all three signs differ here.
NCERT Exemplar:
Matrices
