Solutions Of Linear Algebraic Equations

The matrix is

To find the rank, we can use row operations to reduce it to row echelon form or calculate its determinant. If , the rank is 3. If , then rank is less than 3.

Step 1: Calculate Determinant

Since , the rank is less than 3.

Step 2: Check for Rank 2 (2x2 Submatrix)

Consider the top-left 2x2 submatrix: Its determinant is .

So, a 2x2 submatrix has a non-zero determinant → rank is at least 2.

Step 3: Use Row Operations

This is row echelon form. There are two non-zero rows → rank is:

Step 1: Each entry in a matrix from set is either 0 or 1. So the total number of such matrices is: Step 2: Among these matrices: - Some matrices will have determinant 0 (non-invertible), - Some will have determinant 1 (subset ), - Some will have determinant -1 (subset ).
Step 3: For each matrix with determinant 1, multiplying any one row by -1 flips the sign of the determinant. Thus, for each matrix in , there exists a corresponding matrix in , and vice versa.
Step 4: Therefore, .