Let A, B and C be any matrices of the same order and let k be any scalar. Consider the following statements:
(A) \(A + B = B + A\)
(B) \(A - B = B - A\)
(C) \(k(A + B) = kA + kB\)
(D) If \(kA = O\), then \(A = O\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
Addition and scalar multiplication of matrices are done element by element, so they follow the rules of numbers in each position:
\(a_{ij} + b_{ij} = b_{ij} + a_{ij}\)
\(k(a_{ij} + b_{ij}) = ka_{ij} + kb_{ij}\)
\(a_{ij} - b_{ij} = -(b_{ij} - a_{ij})\)
So \(A + B = B + A\) and \(k(A + B) = kA + kB\) always, while \(A - B = -(B - A)\), and \(kA = O\) also holds for \(k = 0\) with \(A \ne O\).
Checking the statements
(A) \(A + B = B + A\) — Correct. Matrix addition is commutative.
(B) \(A - B = B - A\) — Incorrect. For \(A = I\), \(B = O\): \(A - B = I\) but \(B - A = -I\).
(C) \(k(A + B) = kA + kB\) — Correct. Scalar multiplication distributes over addition.
(D) If \(kA = O\), then \(A = O\). — Incorrect. For \(k = 0\) and \(A = I\), \(kA = O\) but \(A \ne O\).