# Determinants

Mathematics I · Matrices and Vectors · https://tryals.app/learn/mathematics-i/determinants

## What is a Determinant?

The **determinant** is a single number computed from a **square matrix** that captures essential geometric and algebraic properties. For a $2 \times 2$ matrix, it is calculated as:

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

Geometrically, the determinant is the **signed area** scaling factor of the linear transformation. A positive value preserves orientation, a negative value includes a **reflection**, and zero collapses space entirely.

| Determinant Value | Geometric Effect | Invertibility |
| --- | --- | --- |
| $\det \neq 0$ | Scales space, preserves dimension | **Invertible** |
| $\det = 0$ | Squashes space to a lower dimension | **Singular** (no inverse) |

## Core Properties & Pitfalls

The single most important test: **a square matrix is invertible if and only if its determinant is nonzero.** If $\det A = 0$, the matrix is **singular** and $A\mathbf{x} = \mathbf{b}$ lacks a unique solution.

Multiplicative property:
$$\det(AB) = \det(A)\det(B)$$

**Common Pitfalls:**
- Calculating the $2 \times 2$ determinant as $ab - cd$ or $ad + bc$ instead of **$ad - bc$**.
- Believing a zero determinant means invertible. **Zero means singular (not invertible).**
- Swapping any two rows **negates** the determinant, while a zero or repeated row makes it 0.

## Practice questions

3 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. The determinant of a 2×2 matrix is the area of the parallelogram its columns span, and it can be **negative**. What does a negative area mean?

A. That the matrix is not invertible, which negative determinants indicate
B. That the transformation reverses orientation, flipping the plane
C. That the parallelogram is drawn below the x-axis
D. That the area was computed with the columns in the wrong order and should be corrected

**Answer:** B. That the transformation reverses orientation, flipping the plane

**Why:** The magnitude is area and the sign is handedness: a negative determinant means the plane has been flipped, which no amount of rotating and stretching achieves. It is why a mirror image cannot be rotated onto the original, and why orientation matters in graphics for deciding which face of a surface you are seeing.

Page: https://tryals.app/practice/mathematics-i/determinants/the-determinant-of-a-2-2-matrix-is-the-area-of-the-parallelogram-its

### 2. Select every matrix below whose determinant is 0.

A. rows (0, 0) and (5, 7)
B. rows (1, 0) and (0, -1)
C. rows (3, 1) and (1, 3)
D. rows (1, 2) and (2, 4)
E. rows (2, 3) and (4, 6)

**Answer:** A. rows (0, 0) and (5, 7); D. rows (1, 2) and (2, 4); E. rows (2, 3) and (4, 6)

**Why:** Options 1 and 4 have proportional rows and option 1 has a zero row, all forcing determinant 0. Option 2 gives 9 - 1 = 8 and option 2 gives -1, nonzero, hence invertible. A negative determinant is not a zero determinant, which is the trap in the second option.

Page: https://tryals.app/practice/mathematics-i/determinants/select-every-matrix-below-whose-determinant-is-0

### 3. A 2x2 matrix is invertible exactly when ad − bc ≠ 0. Why should a single number decide whether a transformation can be undone?

A. Because the determinant counts the matrix's solutions, and zero solutions means no inverse.
B. Because ad − bc appears in the denominator of the inverse formula, so a zero value is merely a computational obstacle.
C. Because a zero determinant collapses the plane onto a line or point.
D. Because a determinant of zero means the matrix has a row of zeros, which cannot be inverted.

**Answer:** C. Because a zero determinant collapses the plane onto a line or point.

**Why:** The determinant is the area scale factor, so zero means the unit square's image has no area, the plane has been flattened onto a line. Infinitely many points then land on one, and an inverse would have to choose between them with nothing to go on. The formula divides by the determinant *because* of this, not the other way round.

Page: https://tryals.app/practice/mathematics-i/determinants/a-2x2-matrix-is-invertible-exactly-when-ad-bc-0-why-should-a