Let A, B, C, D and E be matrices of order \(2 \times n, 3 \times k, 2 \times p, n \times 3\) and \(p \times k\) respectively. Choose the correct statement(s) from the following?
(A) EB + DB will be defined if \(k = 3, p = n\).
(B) EB + DB will be defined if \(k = 2, p = 3\).
(C) If \(n = p = 2\), then the order of the matrix \(5A^2 - 3C\) is \(2 \times 2\).
(D) If \(n = p\), then the order of the matrix \(5A^2 - 3C\) is \(p \times k\).
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
A product XY is defined when the number of columns of X equals the number of rows of Y; a sum or difference needs two matrices of the same order.
(A) EB + DB will be defined if \(k = 3, p = n\). — Correct. E is \(p \times k\) and B is \(3 \times k\), so EB needs k = 3 and is then \(p \times 3\). D is \(n \times 3\), so DB is \(n \times 3\). With p = n both are \(n \times 3\), and the sum is defined.
(B) EB + DB will be defined if \(k = 2, p = 3\). — Incorrect. With k = 2, E is \(3 \times 2\) and B is \(3 \times 2\); EB would need 2 = 3, so EB is not defined.
(C) If \(n = p = 2\), then the order of the matrix \(5A^2 - 3C\) is \(2 \times 2\). — Correct. A is \(2 \times 2\), so \(A^2\) is \(2 \times 2\); C is also \(2 \times 2\). So \(5A^2 - 3C\) is \(2 \times 2\).
(D) If \(n = p\), then the order of the matrix \(5A^2 - 3C\) is \(p \times k\). — Incorrect. A is \(2 \times n\), so \(A^2\) exists only when n = 2, and then \(5A^2 - 3C\) is \(2 \times 2\), whatever k is. Its order is not \(p \times k\) in general.