Question

If \( A = [a_{ij}] \) is a \( 3 \times 3 \) matrix, where \( a_{ij} = i - 3j \), then which of the following is false?

• A. \( a_{11}<0 \)
• B. \( a_{12} + a_{21} = -6 \)
• C. \( a_{13}>a_{31} \)
• D. \( a_{31} = 0 \)

Hint:

When analyzing matrix elements, compute each element individually and verify the statements step-by-step to ensure accuracy.

Answer 1

The Correct Option is C

Step 1: Define the matrix \( A = [a_{ij}] \).
The elements of the matrix \( A \) are defined by the relation \( a_{ij} = i - 3j \), where \( i \) represents the row index and \( j \) represents the column index.

Step 2: Compute the elements of \( A \).
For a \( 3 \times 3 \) matrix, we compute the elements as follows: \[ a_{11} = 1 - 3(1) = -2, \quad a_{12} = 1 - 3(2) = -5, \quad a_{13} = 1 - 3(3) = -8, \] \[ a_{21} = 2 - 3(1) = -1, \quad a_{22} = 2 - 3(2) = -4, \quad a_{23} = 2 - 3(3) = -7, \] \[ a_{31} = 3 - 3(1) = 0, \quad a_{32} = 3 - 3(2) = -3, \quad a_{33} = 3 - 3(3) = -6. \] Thus, the matrix \( A \) is: \[ A = \begin{bmatrix} -2 & -5 & -8 \\ -1 & -4 & -7 \\ 0 & -3 & -6 \end{bmatrix}. \]
Step 3: Analyze the given options.
- (A) \( a_{11} < 0 \): Since \( a_{11} = -2 \), which is less than zero, this statement is true.

- (B) \( a_{12} + a_{21} = -6 \): Here, \( a_{12} = -5 \) and \( a_{21} = -1 \), so: \[ a_{12} + a_{21} = -5 + (-1) = -6. \] This statement is true.

- (C) \( a_{13} > a_{31} \): Since \( a_{13} = -8 \) and \( a_{31} = 0 \), we have: \[ a_{13} > a_{31} \quad \Rightarrow \quad -8 > 0, \] which is false.

- (D) \( a_{31} = 0 \): As \( a_{31} = 0 \), this statement is true.

Step 4: Conclusion.
The false statement is: \[ \boxed{a_{13} > a_{31}}. \]