Theorem

The determinant of an $n \times n$ matrix $A$ can be computed by a cofactor expansion across any row or down any column. The expansion across the $i$-th row using the cofactors is: $$\text{det} A = a_{i1}C_{i1} + a_{i2}C_{i2} + \cdots + a_{in}C_{in}$$ The cofactor expansion down the $j$-th column is: $$\text{det} A = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj}$$

Theorem

If $A$ is a triangular matrix, then $\text{det} A$ is the product of the entries on the main diagonal of $A$.

Theorem

Let $A$ be a square matrix.

1. If a multiple of one row of $A$ is added to another row to produce a matrix $B$, then $\text{det} B = \text{det} A$.
2. If two rows of $A$ are interchanged to produce $B$, then $\text{det} B = -\text{det} A$.
3. If one row of $A$ is multiplied by $k$ to produce $B$, then $\text{det} B = k \cdot \text{det} A$.

Theorem

A square matrix $A$ is invertible if and only if $\text{det} A \neq 0$.

Theorem

If $A$ is an $n \times n$ matrix, then $\text{det} A^T = \text{det} A$.

Multiplicative Property

If $A$ and $B$ are $n \times n$ matrices, then $\text{det} (AB) = (\text{det} A)(\text{det} B)$.