Question

How do you know if a determinant is zero?

Answer 1

Conditions when the determinant can be zero:

There are three conditions, where the determinant can be zero.

1. If the complete row of a matrix is zero.

Example:

$\left[\begin{array}{ccc}0& 0& 0\\ 1& 1& 2\\ 2& 3& 1\end{array}\right]$etc.

2. If any row or column of a matrix is the constant multiple of another row or column.

Example:

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 4\\ 1& 2& 5\end{array}\right]$etc.

3. If any two rows or columns are equal.

Example:

$\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 2& 5\end{array}\right]$etc.

Hence these are the conditions when the determinant can be zero.