In a third order matrix A, `a_(ij)` denotes the element in the `i^(th)` row and `j^(th)` column.
If `a_(ij) = {{:(0,"for I"= j),(1,"for I" gt j),(-1,"for I"lt j):}`
then the matrix is

Options

A. skew-symmetric
B. symmetric
C. not invertible
D. non- singular

Answer

To solve the problem, we need to construct the third order matrix \( A \) based on the given conditions for its elements \( a_{ij} \).

### Step 1: Define the Matrix Elements
According to the problem statement, the elements of the matrix \( A \) are defined as follows:
- \( a_{ij} = 0 \) if \( i = j \)
- \( a_{ij} = 1 \) if \( i > j \)
- \( a_{ij} = -1 \) if \( i < j \)

### Step 2: Construct the Matrix
Now, we will construct the matrix \( A \) by evaluating the conditions for each element:

1. For \( i = 1 \):
- \( j = 1 \): \( a_{11} = 0 \)
- \( j = 2 \): \( a_{12} = -1 \) (since \( 1 < 2 \))
- \( j = 3 \): \( a_{13} = -1 \) (since \( 1 < 3 \))

2. For \( i = 2 \):
- \( j = 1 \): \( a_{21} = 1 \) (since \( 2 > 1 \))
- \( j = 2 \): \( a_{22} = 0 \)
- \( j = 3 \): \( a_{23} = -1 \) (since \( 2 < 3 \))

3. For \( i = 3 \):
- \( j = 1 \): \( a_{31} = 1 \) (since \( 3 > 1 \))
- \( j = 2 \): \( a_{32} = 1 \) (since \( 3 > 2 \))
- \( j = 3 \): \( a_{33} = 0 \)

Combining all these values, we get the matrix \( A \):

\[ A = \begin{bmatrix} 0 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & 1 & 0 \end{bmatrix} \]

### Step 3: Determine the Properties of the Matrix
Next, we need to check the properties of the matrix \( A \).

#### Check for Skew-Symmetry
A matrix \( A \) is skew-symmetric if \( A^T = -A \).

1. Calculate the transpose \( A^T \):
\[ A^T = \begin{bmatrix} 0 & 1 & 1 \\ -1 & 0 & 1 \\ -1 & -1 & 0 \end{bmatrix} \]

2. Calculate \( -A \):
\[ -A = \begin{bmatrix} 0 & 1 & 1 \\ -1 & 0 & 1 \\ -1 & -1 & 0 \end{bmatrix} \]

Since \( A^T = -A \), the matrix \( A \) is skew-symmetric.

#### Check for Non-Invertibility
To check if the matrix is non-invertible, we can calculate its determinant.

1. Calculate the determinant of \( A \):
\[ \text{det}(A) = 0(-1)(0) + (-1)(-1)(1) + (-1)(1)(1) - (-1)(-1)(1) - 0(1)(1) - (-1)(-1)(0) \]
\[ = 0 + 1 - 1 - 1 + 0 + 0 = -1 \]

Since the determinant is not zero, the matrix is invertible.

### Conclusion
The final properties of the matrix \( A \) are:
- It is skew-symmetric.
- It is invertible.