If $A = [a_{ij}]_{3×2}$, 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
The given matrix $A$ is of size $3 \times 2$ and $a_{ij} = i + j$.
Let's first construct the matrix $A$ using this formula:
$A = \begin{bmatrix} 1 + 1 & 1 + 2 \\ 2 + 1 & 2 + 2 \\ 3 + 1 & 3 + 2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 3 & 4 \\ 4 & 5 \end{bmatrix}$
Option (A): A square matrix must have same number of rows and columns. Since A is $3 \times 2$, it is not a square matrix.
Option (A) is incorrect.
Option (B): We need to compute $a_{21} + a_{32}$.
$a_{21} = 2 + 1 = 3$, $a_{32} = 3 + 2 = 5$, so $a_{21} + a_{32} = 3 + 5 = 8$
Option (B) is correct.
Option (C): A $3 \times 2$ matrix has $3 \times 2 = 6$ elements.
Option (C) is correct.
Option (D): The transpose of a $3 \times 2$ matrix is a $2 \times 3$ matrix:
$A^T = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix}$
The matrix shown in option D is the original matrix A, not its transpose.
Option (D) is incorrect.