Explanation:

The determinant of a matrix is a scalar value that represents certain properties of the matrix. In this case, we are given a matrix A of order 5x5, where each element is either 0 or 1. We need to find the determinant of (A - AT), where AT represents the transpose of matrix A.

Properties of Transpose:
1. The transpose of a matrix is obtained by interchanging its rows and columns.
2. The transpose of a transpose matrix is the original matrix.

Step 1: Finding the Transpose of A (AT)
To find the transpose of matrix A, we need to interchange its rows and columns. Since A is a 5x5 matrix, AT will also be a 5x5 matrix.

Step 2: Subtracting A from AT (A - AT)
To find the matrix (A - AT), we subtract the corresponding elements of A and AT. Since A and AT are both 5x5 matrices, the resulting matrix (A - AT) will also be a 5x5 matrix.

Step 3: Finding the Determinant of (A - AT)
The determinant of a matrix can be calculated using various methods, such as expansion by minors, cofactor expansion, or using row operations to obtain an upper triangular matrix.

However, in this case, we can observe a special property. Since A is a matrix with elements either 0 or 1, the transpose of A (AT) will have the same elements as A. Therefore, when we subtract A from AT, all the corresponding elements will be 0.

Conclusion:
Since all the elements of (A - AT) are 0, the determinant of (A - AT) will also be 0.

(A — A^T) is a skew-symmetric matrix of order 5. Determinant of any skew-symmetric matrix of odd order is zero.