Determinants

Determinant of a $2 \times 2$ matrix $A$: $$\det A = a_{11}a_{22} - a_{12}a_{21}$$

Cofactor expansion #

$A_{ij}$ is the $n \times n$ matrix $A$ without row $i$ and column $j$

$$ C_{ij} = (-1)^{i+j} \det A_{ij} $$ Expansion across the $i$th row: $$ \det A = a_{i1}C_{i1} + a_{i2}C_{i2} + \dots + a_{in}C_{in} $$ Expansion across the $j$th column: $$ \det A = a_{1j}C_{1j} + a_{2j}C_{2j} + \dots + a_{nj}C_{nj} $$

Properties #

• If a multiple of one row is added to another row: $\det B = \det A$
• If two rows are swapped: $\det B = -\det A$
• If one row is multiplied by $k$: $\det B = k \cdot \det A$