# Determinants

Business I · Mathematics I · https://tryals.app/learn/business-i/determinants

## One Number That Judges a Matrix

The **determinant** compresses a square matrix into a single number answering: _does this matrix destroy information?_

**Calculations:**
- **2×2 case**: $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$
- **3×3 case**: Rule of Sarrus (three southeast diagonals minus three northeast).

| $\det A$ | Matrix | System $Ax=b$ |
| --- | --- | --- |
| $\neq 0$ | **Regular**, $A^{-1}$ exists | Exactly one solution |
| $= 0$ | **Singular**, rows dependent | None or infinitely many |

**Key properties**: Swap rows changes sign; $\det(AB) = \det A \cdot \det B$; $\det A^T = \det A$. A row of zeros or equal rows gives $\det = 0$.

> **Tip:** Check $\det A$ before inverting. If zero, save the work; if tiny, be wary of amplified errors.

## The Determinant Is an Area

Geometrically, **$|\det A|$** is the area of the **parallelogram** spanned by the matrix's **column vectors**. In 3D, it scales volumes. The sign shows orientation (negative = flipped).

$$\text{columns aligned} \iff \text{area} = 0 \iff \det = 0$$

**Why singularity destroys information:** When $\det = 0$, the transformation squashes the plane onto a line. Two inputs land on one output, so you **cannot un-squash** it ($A^{-1}$ fails).

> **Common pitfall:** Forgetting absolute value. Area is $|\det A|$; the sign is orientation, not negative area.

## Practice questions

7 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. A transformation matrix has $\det A = 3$. A region of area 8 goes in. Set the slider to the area that comes out.

**Answer:** 24 (within ±1)

**Why:** Every region — squares, blobs, portfolios drawn in risk-return space — has its area multiplied by $|\det| = 3$: output area **24**. One number governs all shapes equally.

Page: https://tryals.app/practice/business-i/determinants/a-transformation-matrix-has-det-a-3-a-region-of-area-8-goes-in

### 2. Complete the fundamental equivalence.

**Answer:** A square matrix is invertible exactly when its determinant is **nonzero**; geometrically, a zero determinant means the unit square collapses to a **line**.

**Why:** Nonzero determinant ⟺ invertible; zero ⟺ collapse to a line (or point), the algebra and the geometry of the same fact.

Page: https://tryals.app/practice/business-i/determinants/complete-the-fundamental-equivalence

### 3. Compute $\det\begin{pmatrix} 5 & 3 \\ 4 & 2 \end{pmatrix}$.

**Answer:** -2

**Why:** $10 - 12 = -2$. Nonzero, so the matrix is invertible; the minus sign says the transformation flips orientation, the parallelogram is traced the other way around.

Page: https://tryals.app/practice/business-i/determinants/compute-detbeginpmatrix-5-3-4-2-endpmatrix

### 4. Sort each fact by what it does to a determinant.

**Answer:**

- Makes it zero: A full row of zeros, Two identical rows, One column is 5× another column
- Changes its sign: Swapping two rows
- Leaves it unchanged: Transposing the matrix

**Why:** Redundancy (zeros, copies, proportionality) kills the determinant; a swap flips orientation; transposition changes nothing, rows and columns tell the same story.

Page: https://tryals.app/practice/business-i/determinants/sort-each-fact-by-what-it-does-to-a-determinant

### 5. Match each statement to its equivalent.

**Answer:**

- $\det A \ne 0$ → $A^{-1}$ exists
- $\det A = 0$ → Rows are linearly dependent
- $|\det A| = 6$ → Areas are scaled by a factor of 6
- $\det A < 0$ → The transformation flips orientation

**Why:** One number, four readings: invertibility, dependence, area scaling, and orientation.

Page: https://tryals.app/practice/business-i/determinants/match-each-statement-to-its-equivalent

### 6. A determinant of zero is often described as 'no unique solution'. What is the geometric fact behind that phrase?

A. The matrix is too large for the number of unknowns
B. The matrix has a row of zeros, making the system unsolvable
C. The equations contradict each other, so no solution exists at all
D. The matrix collapses space onto a lower dimension

**Answer:** D. The matrix collapses space onto a lower dimension

**Why:** Zero determinant means the transformation flattens the space, and flattening destroys the information you would need to reverse it. Option 3 is one possible consequence but not the meaning, a singular system may have infinitely many solutions rather than none.

Page: https://tryals.app/practice/business-i/determinants/a-determinant-of-zero-is-often-described-as-no-unique-solution

### 7. An economist’s two policy instruments act on two targets through matrix $A$, and $\det A = 0$. What does this mean for policy?

A. The determinant must be recomputed in different units
B. Both instruments are twice as powerful as expected
C. The model has a unique solution for every target
D. The instruments are redundant; some targets are unreachable

**Answer:** D. The instruments are redundant; some targets are unreachable

**Why:** Zero determinant means the instrument columns are dependent: pulling lever 2 is a disguised multiple of pulling lever 1. The plane of targets collapses to a line, everything off that line is unreachable at any policy strength.

Page: https://tryals.app/practice/business-i/determinants/an-economists-two-policy-instruments-act-on-two-targets-through