Question

How is the value of the determinant related to whether a matrix is singular or non-singular?

Answer 1

Question 1. How is the value of the determinant related to whether a matrix is singular or non-singular?

Solution. By definition a square matrix is called singular if it is not invertible, i. e. if it does not have the inverse matrix. There is a criterion of being singular in terms of determinant: a matrix is singular if and only if its determinant is zero. For example, the determinant of

$A=\begin{pmatrix}1&2\cr 2&5\end{pmatrix}$

equals 1, so $A$ should be nonsingular. Indeed, one can easily see that

$A^{-1}=\begin{pmatrix}5&-2\cr-2&1\end{pmatrix}.$

But, for instance

$B=\begin{pmatrix}1&2\cr 2&4\end{pmatrix}$

is singular, because its determinant is zero.

Answer: a matrix in singular exactly when its determinant is zero.