If all the elements in a square matrix A of order 3 are equal to 1 or - 1, then `|A|`, is

Options

A. an odd number
B. an even number
C. an imaginary number
D. a real number

Answer

To find the determinant of a square matrix \( A \) of order 3 where all elements are either 1 or -1, we can analyze the situation step by step.

### Step 1: Define the Matrix
Let's define a general 3x3 matrix \( A \) where each element is either 1 or -1. For example, we can write:
\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]
where each \( a_{ij} \) (for \( i, j = 1, 2, 3 \)) can be either 1 or -1.

### Step 2: Calculate the Determinant
The determinant of a 3x3 matrix can be calculated using the formula:
\[ |A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

### Step 3: Consider Different Cases
1. **Case 1: All elements are 1**
\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \]
The determinant of this matrix is 0 because all rows are identical.

2. **Case 2: Two elements are 1 and one is -1 in each row**
For example:
\[ A = \begin{pmatrix} 1 & 1 & -1 \\ 1 & -1 & 1 \\ -1 & 1 & 1 \end{pmatrix} \]
The determinant of this matrix can also be calculated and will yield 0 due to the linear dependence of the rows.

3. **Case 3: One element is 1 and two are -1 in each row**
For example:
\[ A = \begin{pmatrix} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{pmatrix} \]
Again, the determinant will be 0 due to the same reasoning.

### Conclusion
In all cases considered, whether all elements are 1, or a mix of 1 and -1, the determinant \( |A| \) equals 0. Thus, we conclude that:
\[ |A| = 0 \]