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:

Given $A = [a_{ij}]_{3 \times 2}$ where $a_{ij} = i + j$

The notation $3 \times 2$ means A has 3 rows and 2 columns.

The formula $a_{ij} = i + j$ gives each element by adding the row number and column number.

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Computing all elements using $a_{ij} = i + j$:

$a_{11} = 1 + 1 = 2$

$a_{12} = 1 + 2 = 3$

$a_{21} = 2 + 1 = 3$

$a_{22} = 2 + 2 = 4$

$a_{31} = 3 + 1 = 4$

$a_{32} = 3 + 2 = 5$

Therefore: $A = \begin{bmatrix}2 & 3 \\ 3 & 4 \\ 4 & 5\end{bmatrix}$

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Checking option (A): A is a square matrix

A square matrix has equal rows and columns.

Matrix A is $3 \times 2$ with 3 rows and 2 columns.

Since $3 \neq 2$, option (A) is false.

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Checking option (B): $a_{21} + a_{32} = 8$

$a_{21} = 2 + 1 = 3$

$a_{32} = 3 + 2 = 5$

$a_{21} + a_{32} = 3 + 5 = 8$

Option (B) is true.

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Checking option (C): Number of elements in A is 6

Total elements $= \text{rows} \times \text{columns}$

Total elements $= 3 \times 2$

Total elements $= 6$

Option (C) is true.

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Checking option (D): Transpose of $A = \begin{bmatrix}2 & 3 \\ 3 & 4 \\ 4 & 5\end{bmatrix}$

The matrix shown is A itself, not its transpose.

The transpose swaps rows and columns.

$A^T = \begin{bmatrix}2 & 3 & 4 \\ 3 & 4 & 5\end{bmatrix}$

This is a $2 \times 3$ matrix.

Option (D) is false.

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Only statements (B) and (C) are correct.