Determinant

Determinant

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Prerequisites

Basis Vector
Span
Linear independent/dependent

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What is Determinant

1. What is Determinant?

A matrix represents a linear transformation.

The determinant tells you How much area (2D) is scaled or Whether the transformation flips orientation

\[|\det(A)| = \text{area or volume scale factor}\]

• Stretching → magnitude increases
• Squishing → magnitude decreases
• Collapsing → determinant zero

2. Orientation of determinant.

\[\text{sign of } \det(A) = \text{orientation}\]

Determinant is NOT absolute are, but signed area.

• If $A\hat{j}$ is to the left of $A\hat{i}$ → orientation preserved → positive
• If $A\hat{j}$ is to the right of $A\hat{i}$ → orientation flipped → negative

So:

\[\det(A) > 0 \Rightarrow \text{no flip}\] \[\det(A) < 0 \Rightarrow \text{reflection occurred}\]

• Flipping → sign changes

3. 2D Matrix Interpret

3-1. Scale of Determinant

Take a $2 \times 2$ matrix:

\[A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

where basis vector:

\[A\hat{i} = \begin{bmatrix} a \\ c \end{bmatrix}, \qquad A\hat{j} = \begin{bmatrix} b \\ d \end{bmatrix}\]

• Left column → where $\hat{i}$ lands
• Right column → where $\hat{j}$ lands

A unit square becomes a parallelogram formed by these two vectors.

The determinant equals the signed area of that parallelogram:

\[\det(A) = ad - bc\]

If $\det(A) = 0$, space collapses to a line

3-2. Area shape of Determinant

• $a$ affects stretching along x-direction
• $d$ affects stretching along y-direction
• $b$ and $c$ introduce shearing / diagonal distortion

Case 1: $b = 0$ and $c = 0$

\[A = \begin{bmatrix} a & 0 \\ 0 & d \end{bmatrix}\]

Just x-direction stretched by $a$ and y-direction stretched by $d$. Pure rectangle → scaled rectangle.

Case 2: $b$ and $c$ are not zero

Then the square becomes a tilted parallelogram.

• $b$ moves the x-basis in y-direction
• $c$ moves the y-basis in x-direction

They introduce diagonal stretching / shearing. So the cross-interaction term $bc$ reduces (or increases) area depending on orientation.

4. 3x3 Determinant

\[det(A)= \det \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\] \[\det(A) = a(ei-fh)+b(fg-di)+c(dh-eg)\]

5. NxN Determinant

\[det = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{23} & C_{33} \end{bmatrix}\] \[\det(A) = a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}\]

by Row: \(\det(A) = \sum_{j=1}^{N} a_{ij} (-1)^{i+j} \det(M_{ij})\)

by Column: \(\det(A) = \sum_{i=1}^{N} a_{ij} (-1)^{i+j} \det(M_{ij})\)