Let B, C and D be matrices of orders \(2 \times q\), \(m \times 2\) and \(r \times q\) respectively. The restrictions on m, q and r so that \(DB + CB\) is defined are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(q = 2\) and \(r = m\)
A product is defined only when the number of columns of the first matrix equals the number of rows of the second.
\(CB\): \((m \times 2)(2 \times q)\) is defined, of order \(m \times q\)
\(DB\): \((r \times q)(2 \times q)\) needs \(q = 2\), and is then of order \(r \times 2\)
With \(q = 2\), CB is of order \(m \times 2\). Two matrices can be added only when they have the same order:
\(r \times 2 = m \times 2\), so \(r = m\)
Hence \(q = 2\) and \(r = m\).
Why the other options are incorrect
Option (2) \(q = m\) and \(r = 2\) — Incorrect. For \(m = q = 3\), \(r = 2\), the product DB is \((2 \times 3)(2 \times 3)\), which is not defined.
Option (3) \(q = 2\), with no restriction on m and r — Incorrect. For \(q = 2\), \(m = 3\), \(r = 1\), DB is \(1 \times 2\) and CB is \(3 \times 2\), so they cannot be added.
Option (4) \(m = 2\) and \(r = q\) — Incorrect. For \(m = 2\), \(r = q = 3\), DB is \((3 \times 3)(2 \times 3)\), which is not defined.