Question

Under what condition(s) does a square matrix have no inverse?

Answer 1

I'm thinking if the determinant is 0, there is no inverse, but I'm not sure if I'm right or there are other answers.

Answer 2

I'd also choose YOUR answer: If the det. is 0, the matr. has no inverse.

Answer 3

Oh, okay, thank you!

Answer 4

@Sushi121212
Exactly, if the determinant of a matrix is zero, the inverse of the matrix does not exist.
The converse of the statement is also true, i.e.
if the inverse of the matrix does not exist, the determinant of the matrix is zero.
Which means that
A matrix is not invertible if and only if the determinant is zero.

Also, it may be helpful to know some properties of matrices which have a determinant of zero:
1. When there are identical rows or identical columns.
Example: following matrix has a determinant of zero (first row equals last row)
5 2 4 1
4 2 9 8
2 0 3 7
5 2 4 1
2. More generally, when two rows or two columns are linearly dependent, i.e. if one row of the matrix equals another row multiplied by a constant.
Example: the following matrix has a determinant of zero (row 2 = row 3 *2 )
5 2 4 1
4 0 6 2
2 0 3 1
7 3 6 0
3. When a row or a column has all zeroes, the determinant is zero.
Example: following matrix has a zero determinant (column 2 is all zeroes)
5 0 4 1
4 0 6 2
2 0 9 8
7 0 6 0

Answer 5

Oh wow!! Thank you so much for taking the time to write all that!! I have to give you a medal someway. I already gave one away. I'll go and give you a medal on another question that you answered. Thank you so much! I really appreciate it!