A matrix is nothing but a collection of numbers arranged in rectangular boxes. Hence a 1 x 1 matrix having only one row and one column is nothing but the scalar/number itself and the determinant is equal to the value of the scalar.
Example of a 1 x 1 matrix with positive determinant:
Consider the matrix shown below.
The above matrix has determinant equal to 7. Since the determinant is nonzero the equation Ax = 0 must have only the trivial solution. This is clearly true since the equation 7x = 0 \text{ has } x = 0 as the only solution. This also gives us some motivation for why the determinant of a 1 x 1 matrix is defined in the manner above.
Example of a 1 x 1 matrix with determinant zero:
The matrix shown below clearly has determinant equal to zero.
This means that the equation Ax = 0 must have a non-trivial solution. This is clearly true since the equation 0x = 0 is true for any value of x belonging to real numbers.
