If \(A = [a_{ij}]_{3 \times 2}\), where \(a_{ij} = i + j\), then
(A) A is a square matrix
(B) \(a_{21} + a_{32} = 8\)
(C) Number of elements in A is 6
(D) Transpose of A = \(\begin{bmatrix}2 & 3\\3 & 4\\4 & 5\end{bmatrix}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (C) only
A has 3 rows and 2 columns, and each entry is the sum of its row and column numbers:
\(\displaystyle A = \begin{bmatrix}2 & 3\\3 & 4\\4 & 5\end{bmatrix}\)
\(a_{21} = 2 + 1 = 3,\quad a_{32} = 3 + 2 = 5\)
The transpose interchanges rows and columns, so it has order 2 × 3:
\(\displaystyle A^T = \begin{bmatrix}2 & 3 & 4\\3 & 4 & 5\end{bmatrix}\)
So A is a 3 × 2 matrix with 6 elements, \(a_{21} + a_{32} = 3 + 5 = 8\), and \(A^T\) is of order 2 × 3.
Checking the statements
(A) A is a square matrix — Incorrect. A has 3 rows but only 2 columns.
(B) \(a_{21} + a_{32} = 8\) — Correct. \(3 + 5 = 8\).
(C) Number of elements in A is 6 — Correct. \(3 \times 2 = 6\).
(D) Transpose of A = \(\begin{bmatrix}2 & 3\\3 & 4\\4 & 5\end{bmatrix}\) — Incorrect. This is A itself; \(A^T\) is of order 2 × 3.