The geometric meaning of a determinant is useful, but practical work requires ways to calculate it from matrix entries.

Two-by-two matrices

For

$$A=\begin{pmatrix}a&b\c&d\end{pmatrix},$$

the determinant is

$$\det A=ad-bc.$$

For example,

$$\det\begin{pmatrix}2&1\3&4\end{pmatrix}=2\cdot4-1\cdot3=5.$$

Because the result is nonzero, this matrix is invertible.

Cofactor expansion

For larger matrices, a determinant can be expanded along any row or column. If row $i$ is chosen,

$$\det A=\sum_j (-1)^{i+j}A_{ij}\det M_{ij},$$

where $M_{ij}$ is obtained by deleting row $i$ and column $j$.

Choosing a row or column containing several zeros can make the calculation much shorter.

Row operations and determinants

Elementary row operations change determinants in predictable ways:

• swapping two rows changes the sign;
• multiplying a row by $c$ multiplies the determinant by $c$;
• adding a multiple of one row to another leaves the determinant unchanged.

This allows a matrix to be simplified toward triangular form while tracking the determinant.

Triangular matrices

For a triangular matrix, the determinant is the product of its diagonal entries:

$$\det A=A_{11}A_{22}\cdots A_{nn}.$$

Together, cofactor expansion and row-operation properties turn determinant calculation into a structured procedure rather than a single large formula.