Let the eigenvalues of a matrix of order 3 be different from each other. If the determinant is used, then the rank of a is

Let the eigenvalues of a matrix of order 3 be different from each other. If the determinant is used, then the rank of a is

What about determinants?
If the determinant is not zero, then the rank of a is 3
If there is no condition that the determinant is not zero, the rank of a should be greater than or equal to 2

Linear Algebra: when finding the eigenvalue of matrix A, can we subtract the identity matrix from the matrix after finding the rank of the matrix, and then solve the determinant? Is the result the same
Linear Algebra: when calculating the eigenvalue of matrix A, can we subtract the identity matrix from the matrix after the rank of the matrix, and then solve the determinant? Is the result the same? Explain the reason,

No, the transformation used to calculate rank will change the value of determinant, unless you also perform the same elementary transformation on the unit matrix, and then use the transformed matrix instead of the unit matrix to subtract the rank matrix

Is it necessary to calculate the determinant nonzero first and then prove the invertibility to find the inverse of the second order matrix

Yes
And the formula can be used
A =
a b
c d
|A| = ad-bc
A^-1 = (1/|A|) *
d -b
-c a
That is, the main diagonal exchange position, sub diagonal negative sign