Animated Solution for Mathematics - Matrices and Determinants: The number of $3\times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$, is

Imagine a $3\times 3$ grid, a blank canvas of nine empty slots. We are tasked with placing exactly four $1$s and five $0$s into this grid to form a matrix $A$.

The core constraint is that the matrix must be non-singular, meaning its determinant must satisfy:

To ensure $|A| eq 0$, we must avoid the "Zero Trap." If any row or any column is entirely filled with zeros, the determinant is guaranteed to be zero.

Therefore, to maintain a non-zero determinant, every single row and every single column must contain at least one $1$. This is our golden rule. If any row or column is left empty, the matrix is singular, and the configuration is invalid.

Let us begin with the most structured configuration: the Identity matrix $I$. With $1$s on the main diagonal and $0$s elsewhere, we have $|A|=1$.

In this configuration, we have used three $1$s. We have one $1$ remaining to place in any of the six available $0$ positions.

If we place this fourth $1$ in any of these six positions, the determinant remains non-zero. For example, placing the fourth $1$ at position $(1,2)$ results in a determinant of $1$. Thus, we have identified six distinct non-singular matrices derived from this base.

The Identity matrix is not the only valid structure. We can also utilize permutation matrices. Consider a matrix with $1$s at positions $(1,2)$, $(2,1)$, and $(3,3)$.

The determinant of this matrix is $-1$, which is non-zero. This serves as another valid base.

If we add our fourth $1$ to this base, such as at position $(1,1)$, the determinant remains non-zero. This confirms the existence of at least one additional matrix, bringing our count to at least seven.

By exploring these structures, we see that the number of such matrices is not limited to the initial six. We have found six from the Identity base and at least one more from the permutation base.

The logical conclusion, supported by the problem constraints, is that the number of such matrices is at least 7. This journey through the grid demonstrates that looking beyond the obvious base case reveals a much richer, more complex mathematical reality.