Let \(A\) be an \(n \times n\) matrix with entries from a field.
Recall that the determinant of \(A\),
denoted by \(\det(A)\), is defined to be
\[\sum_{\sigma \in S_n} (-1)^{\text{#inv}(\sigma)}
A_{1,\sigma(1)}\cdots A_{n,\sigma(n)}.\]

Here are some properties of the determinant.

\(\det(A^\mathsf{T}) = \det(A).\)

If \(A\) is invertible,
then \(\displaystyle\det(A^{-1}) = \frac{1}{\det(A)}.\)

Where \(\alpha\) is a scalar,
\(\det(\alpha A) = \alpha^n\det(A).\)

\(\det(A) \neq 0\) if and only if \(A\) is nonsingular.

Examples

Let \(A = \begin{bmatrix} 1 & 2 \\ -1 & 1\end{bmatrix}\).
Note that \(\det(A) = 3\).

\( \det(A^\mathsf{T}) = \det(A) = 3.\)

Since \(\det(A)\neq 0\), \(A\) is nonsingular and therefore invertible.
Hence, \( \det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{3}.\)

\( \det(2A) = 2^2\det(A) = 4\cdot 3 = 12\) since \(A\) is \(2\times 2\).

\( \det(-A) = \det((-1)A) = (-1)^2\det(A) = \det(A) = 3\). Notice that
in this case, \(\det(-A) = \det(A)\)!

\(\det(A) \neq 0\) iff \(A\) is nonsingular

We now give a proof of this important result.

Let \(R\) be the RREF of \(A\). Then there exist elementary matrices
\(M_1,\ldots,M_k\) such that \(M_{k} M_{k-1} \cdots M_1 A = R\).
Using the fact that the determinant of a product of square
matrices is the same as the product of the determinants of the matrices,
we get \[\det(M_k) \det(M_{k-1})\cdots \det(M_1) \det(A) = \det(R).\]

As \(R\) is a square matrix, \(A\) is nonsingular iff \(R\) does not
contain a row of 0's.
If \(R\) does contain a row of 0's, then \(\det(R) = 0\).
Otherwise, \(R\) must be the identity matrix with determinant 1.

Hence, to complete the proof, it suffices to show that \(\det(M_i) \neq 0\)
for all \(i = 1,\ldots,k\). Note that every elementary matrix
is either a triangular matrix with nonzeros on the diagonal,
or a permutation matrix which has nonzero determinant.
Hence, \(\det(M_i) \neq 0\) for all \(i = 1,\ldots, k\) as desired.

With this result, we can state that
if \(A \in \mathbb{F}^{n\times n}\) for some
field \(\mathbb{F}\), then the following statements are equivalent: