## Basic properties

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:

• $$A$$ is invertible.

• $$A$$ is nonsingular.

• $$\det(A) \neq 0$$.

## Exercises

1. Compute the determinants of each of the following matrices:

1. $$2\begin{bmatrix} 2 & 3 \\ 0 & 2\end{bmatrix}$$

2. $$\begin{bmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f\end{bmatrix}^\mathsf{T}$$

2. Prove that if $$A$$ is invertible, then $$\displaystyle\det(A^{-1}) = \frac{1}{\det(A)}.$$