Skill: Interpret the determinant as a test of invertibility

For a square matrix, calculate its determinant to test whether it is invertible:

• If $\det(A)\ne 0$, then $A$ is invertible. Its rows and columns are independent, and the system $A\mathbf{x}=\mathbf{b}$ has exactly one solution for every $\mathbf{b}$.
• If $\det(A)=0$, then $A$ is singular. Its rows or columns are dependent, so the transformation collapses area and loses information.

Worked example

Determine whether

$$A=\begin{pmatrix}2&1\\3&2\end{pmatrix}$$

is invertible.

For a $2\times2$ matrix, subtract the product of the bottom-left and top-right entries from the product of the diagonal entries:

$$\det(A)=(2)(2)-(1)(3)=4-3=1.$$

Since $\det(A)=1\ne 0$, the matrix is invertible. Its transformation does not collapse area, and any system using this matrix as its coefficient matrix has a unique solution for each right-hand side. For instance,

$$\begin{pmatrix}2&1\\3&2\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}5\\8\end{pmatrix}$$

has the unique solution $x=2,\ y=1$.