The Determinant of a Matrix

# The Determinant of a Matrix

Recall from The Invertibility of a Matrix page that a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is invertible if and only if $ad - bc \neq 0$. This value, $ad - bc$ is known as the Determinant for the $2 \times 2$ matrix $A$ which we define formally below.

 Definition: The Determinant of the $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ denoted $\det (A)$ is $\det (A) = ad -bc$.

With this definition, we can reword the theorem from The Invertibility of a Matrix page to be that a $2 \times 2$ matrix $A$ as defined above is invertible if and only if $\det (A) \neq 0$.

Of course, the determinant quantity is defined for larger square matrices, and once again, you're invited to read much more detail on the Linear Algebra page on Math Online into computing the determinants of $n \times n$ matrices. We thus generate the following equivalent statements regarding the invertibility of a matrix.

 Theorem 1: Let $A$ be an $n \times n$ matrix. Then the following statements are equivalent: a) $A$ is invertible. b) $\det A \neq 0$. c) There exists a unique solution to the system $Ax = b$. d) The trivial solution is the only solution to $Ax = 0$.

The proof that $a \implies b$ requires the reader to have knowledge about cofactor and adjoint matrices. We saw that for $2 \times 2$ matrices that:

(1)
\begin{align} \quad A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & - b \\ -c & a \end{bmatrix} = \frac{1}{\det (A)} \begin{bmatrix} d & - b \\ -c & a \end{bmatrix} \end{align}

More generally, the matrix $\begin{bmatrix} d & - b \\ -c & a \end{bmatrix}$ is known as the Adjoint Matrix of $A$. For matrices of size larger than $2 \times 2$, we can also define an adjoint matrix and show that for any $n \times n$ matrix $A$, that:

(2)
So $A^{-1}$ exists if $\det (A) \neq 0$.