What does it mean to have a determinant equal to zero?

After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$.

I hope someone can explain this to me in plain English.

Solution 1:

For an $n\times n$ matrix, each of the following is equivalent to the condition of the matrix having determinant $0$:

• The columns of the matrix are dependent vectors in $\mathbb R^n$

• The rows of the matrix are dependent vectors in $\mathbb R^n$

• The matrix is not invertible.

• The volume of the parallelepiped determined by the column vectors of the matrix is $0$.

• The volume of the parallelepiped determined by the row vectors of the matrix is $0$.

• The system of homogenous linear equations represented by the matrix has a non-trivial solution.

• The determinant of the linear transformation determined by the matrix is $0$.

• The free coefficient in the characteristic polynomial of the matrix is $0$.

Depending on the definition of the determinant you saw, proving each equivalence can be more or less hard.

Solution 2:

For me, this is the most intuitive video on the web that explains determinants, and everyone who wants a deep and visual understanding of this topic should watch it:

The determinant by 3Blue1Brown

The whole playlist is available at this link:

Essence of linear algebra by 3Blue1Brown

The crucial part of the series is "Linear transformations and matrices". If you understand that well, everything else will be like a piece of cake. Literally: plain English + visual.

Solution 3:

If the determinant of a square matrix $n\times n$ $A$ is zero, then $A$ is not invertible. This is a crucial test that helps determine whether a square matrix is invertible, i.e., if the matrix has an inverse. When it does have an inverse, it allows us to find a unique solution, e.g., to the equation $Ax = b$ given some vector $b$.

When the determinant of a matrix is zero, the system of equations associated with it is linearly dependent; that is, if the determinant of a matrix is zero, at least one row of such a matrix is a scalar multiple of another.

[When the determinant of a matrix is nonzero, the linear system it represents is linearly independent.]

When the determinant of a matrix is zero, its rows are linearly dependent vectors, and its columns are linearly dependent vectors.