Question

What if the determinant is zero?

Answer 1

If the determinant is zero:

• The matrix of the determinant is non-singular and not invertible.
• The matrix of the determinant may be a zero matrix.
• The system of equations associated with the matrix is linearly dependent.
• The rows and columns of the matrix of the determinant are linearly dependent vectors.

Example:

$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 0& 2\\ 0& 5& 5\end{array}\right]$

The determinant of $A$ is,

$\left|A\right|=1\left(0-10\right)-2\left(10-0\right)+3\left(10-0\right)$

$\Rightarrow$ $\left|A\right|=-10-20+30$

$\Rightarrow$ $\left|A\right|=0$

Hence,possible results are explained if determinant is zero.