# Determinants

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/determinants

## Determinants & Formulas

The **determinant** is a scalar value encoding a square matrix's invertibility, volume scaling, and orientation.

For a $2 \times 2$ matrix:
$$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$

For $3 \times 3$, use **cofactor expansion** along the first row:
$$\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$
where $C_{ij} = (-1)^{i+j}M_{ij}$ and $M_{ij}$ is the minor.

**Key properties:**
| Operation | Rule |
| --- | --- |
| Product | $\det(AB) = \det(A)\det(B)$ |
| Transpose | $\det(A^T) = \det(A)$ |
| Scaling | $\det(\alpha A) = \alpha^n \det(A)$ |
| Invertibility | **Invertible** $\iff \det(A) \neq 0$ |

## Geometry & Pitfalls

Geometrically, absolute determinants measure space scaling:
| Space | Geometric Meaning |
| --- | --- |
| $\mathbb{R}^{2}$ | Parallelogram area |
| $\mathbb{R}^{3}$ | Parallelepiped volume |

A **negative** determinant means the transformation **reverses orientation**.

**Physics link:** The Jacobian determinant $\det(\partial x_i/\partial q_j)$ handles coordinate changes in integrals.

> **Common pitfall:** $\det(A+B) \neq \det A + \det B$. Determinants respect multiplication, but never addition.

## Practice questions

6 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Match each determinant property to its statement.

**Answer:**

- det(AB) → $det(A)$ · $det(B)$
- det(Aᵀ) → det(A)
- $det(cA)$ for n$\times n$ → cⁿ det(A)
- $det(A^{-1})$ → 1/det(A)

**Why:** These properties make determinant calculations more efficient.

Page: https://tryals.app/practice/physics-i/determinants/match-each-determinant-property-to-its-statement

### 2. Which statements are true when det(A) = 0?

A. A is the zero matrix
B. A is singular
C. A has no inverse
D. The columns of A are linearly dependent

**Answer:** B. A is singular; C. A has no inverse; D. The columns of A are linearly dependent

**Why:** det(A) = 0 means singular, but A need not be the zero matrix (e.g., [[1,2],[2,4]]).

Page: https://tryals.app/practice/physics-i/determinants/which-statements-are-true-when-det-a-0

### 3. Sort these determinant values into Invertible or Singular.

**Answer:**

- Invertible: det = 5, $det = -3$, det = 0.001
- Singular: det = 0, det = 0.0

**Why:** Any nonzero determinant, however small, means the matrix is invertible.

Page: https://tryals.app/practice/physics-i/determinants/sort-these-determinant-values-into-invertible-or-singular

### 4. A 2×2 matrix $M$ has $\det M = 0$. It transforms the plane. What does it do to the unit square?

A. Stretches it to area 0.5
B. Flattens it onto a line
C. Reflects it, flipping orientation
D. Rotates it without changing area

**Answer:** B. Flattens it onto a line

**Why:** Zero determinant means total collapse: the square is squashed flat, dimensions are lost, and distinct points pile onto each other. That is exactly why the matrix cannot be inverted, you cannot un-flatten a pancake back into its original points.

Page: https://tryals.app/practice/physics-i/determinants/a-2-2-matrix-m-has-det-m-0-it-transforms-the-plane-what-does-it

### 5. The determinant’s absolute value is the factor by which a transformation scales area or volume. What does a determinant of zero mean geometrically?

A. That the matrix has no eigenvalues
B. That the transformation is a pure rotation
C. That it collapses space onto a lower dimension
D. That the transformation leaves all areas unchanged

**Answer:** C. That it collapses space onto a lower dimension

**Why:** Zero volume means dimensions were lost, and lost information cannot be restored. Singularity and non-invertibility are the same fact stated algebraically and geometrically.

Page: https://tryals.app/practice/physics-i/determinants/the-determinants-absolute-value-is-the-factor-by-which-a

### 6. If det(A) = 0, then A is invertible.

**Answer:** False

**Why:** A matrix is invertible if and only if its determinant is nonzero.

Page: https://tryals.app/practice/physics-i/determinants/if-det-a-0-then-a-is-invertible