Determinants

Definition for $2 \times 2$

$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

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Definition for $3 \times 3$

Using cofactor expansion:

$\det(A) = a_{11}M_{11} - a_{12}M_{12} + a_{13}M_{13}$

where $M_{ij}$ is the minor (determinant of $2 \times 2$ matrix after removing row $i$ and column $j$).

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Cofactor

The cofactor $C_{ij}$ is:

$C_{ij} = (-1)^{i+j} M_{ij}$

Pattern of signs: $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$

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General Method

The determinant can be computed by expansion along any row or column:

$\det(A) = \sum_{j=1}^{n} a_{ij}C_{ij} \quad \text{(expansion along row } i)$

$\det(A) = \sum_{i=1}^{n} a_{ij}C_{ij} \quad \text{(expansion along column } j)$

Strategy: Expand along a row/column with many zeros.

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Levi-Civita Symbol

For $n = 3$, define $\epsilon_{ijk}$:

$\epsilon_{ijk} = \begin{cases} +1 & (i,j,k) \text{ even permutation of } (1,2,3) \\ -1 & (i,j,k) \text{ odd permutation of } (1,2,3) \\ 0 & \text{any index repeated} \end{cases}$

Determinant:

$\det(A) = \sum_{i,j,k=1}^{3} \epsilon_{ijk} a_{1i} a_{2j} a_{3k}$

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Properties

Property	Formula
Transpose	$\det(A^T) = \det(A)$
Product	$\det(AB) = \det(A)\det(B)$
Scalar	$\det(\lambda A) = \lambda^n\det(A)$ for $n \times n$
Inverse	$\det(A^{-1}) = 1/\det(A)$
Triangular	$\det = \text{product of diagonal elements}$

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Row Operations

Operation	Effect on determinant
Swap two rows	Multiply by $-1$
Multiply row by $\lambda$	Multiply by $\lambda$
Add multiple of one row to another	No change

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Geometric Interpretation

$|\det(A)|$ = volume of parallelepiped formed by columns of $A$.

$2 \times 2$: Area of parallelogram.

$3 \times 3$: Volume of parallelepiped.

$\det(A) = 0$ means columns are linearly dependent (volume collapses).

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Example

$\det\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{pmatrix}$

Expand along first row:

$= 1\det\begin{pmatrix} 5 & 6 \\ 8 & 10 \end{pmatrix} - 2\det\begin{pmatrix} 4 & 6 \\ 7 & 10 \end{pmatrix} + 3\det\begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix}$

$= 1(50 - 48) - 2(40 - 42) + 3(32 - 35)$

$= 2 + 4 - 9 = -3$

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Determinant Test for Linear Independence

$n$ vectors in $\mathbb{R}^n$ are linearly independent $\Leftrightarrow$ determinant of the matrix formed by them is non-zero.

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Why Determinants Matter

• Determine if linear system has unique solution
• Compute matrix inverse
• Calculate eigenvalues (characteristic polynomial)
• Measure volume change under transformation
• Test for singular matrices