AcademyDeterminants
Academy
Determinants And Invertibility
Level 1 - Linear Algebra topic page in Determinants.
Principle
The determinant is the main scalar test for invertibility. A square matrix is invertible exactly when its determinant is non-zero.
This connects determinant calculations to linear systems: a non-zero determinant means every right-hand side has a unique solution.
Notation
\(A\)
an n by n matrix
\(A^{-1}\)
inverse of A
\(\det A\)
determinant of A
\(A\mathbf x=\mathbf b\)
a square linear system
\(\mathbf 0\)
the zero vector
The Core Test
The determinant invertibility test is
Determinant invertibility test
\[A\text{ is invertible}\Longleftrightarrow \det A\ne0\]
Equivalently,
Singular test
\[A\text{ is singular}\Longleftrightarrow \det A=0\]
Consequences for systems:
- If \(\det A\ne0\), then \(A\mathbf x=\mathbf b\) has exactly one solution for every \(\mathbf b\).
- If \(\det A=0\), then \(A\mathbf x=\mathbf 0\) has a non-zero solution.
Worked Cases
Question
Decide whether \(A=\begin{pmatrix}2&5\\1&3\end{pmatrix}\) is invertible.
Answer
Compute the determinant: \(\det A=2\cdot3-5\cdot1=6-5=1\). Since \(\det A\ne0\), the matrix is invertible. Therefore \(A\mathbf x=\mathbf b\) has a unique solution for every \(\mathbf b\in\mathbb R^2\).
Examples
Question
If \(\det A=0\), what can be said about \(A\mathbf x=\mathbf 0\)?
Answer
A zero determinant means \(A\) is not invertible. For a square matrix, this is equivalent to saying that the homogeneous system \(A\mathbf x=\mathbf 0\) has at least one non-zero solution. So the solution set contains more than just \(\mathbf 0\).