Question

Let \( A = [a_{ij}] \) be a square matrix of order 3 such that \( a_{ij} = \hat{i} - 2 \hat{j} \). Then, which of the following is true?

Hint:

To solve matrix-related problems, evaluate the individual elements and ensure the conditions match before confirming the answer.

Answer 1

We are given the square matrix \( A = [a_{ij}] \) of order 3 such that: \[ a_{ij} = \hat{i} - 2 \hat{j}. \] This means: - The first row is \( a_{11}, a_{12}, a_{13} \) with each element having values \( a_{ij} = 1 - 2 \). - Similarly, the second and third rows follow the same form.
Now, evaluate the options:
- \( a_{12}>0 \): False, since \( a_{12} = 1 - 2 = -1 \).
- \( \text{all} \ a_{ij}<0 \): False, since some elements are positive.
- (C) \( a_{13} + a_{31} = -6 \): False, based on the matrix values.
- (D) \( a_{23}>a_{32} \): True, based on the given matrix structure.
Thus, the correct answer is \( \boxed{D} \).