Determinants
Before The determinant is a test for invertibility, so it builds on inverses and on the Invertible Matrix Theorem. Builds on: Matrix Operations and Invertibility · The Invertible Matrix Theorem
Next Eigenvalues are found by solving the equation det(A - λI) = 0. Leads to: Eigenvalues and Eigenvectors
Before You Start: Prerequisite Check
Can you do these? (Click to reveal self-test)
Test yourself on these prerequisite skills:
Product formula: Is $\det(AB) = \det(A)\det(B)$ always true? (Predict yes or no before reading on.)
Check
Yes, this is a key property of determinants proved in Lay Section 3.2. It is one of the major tools for computing determinants of products without expanding.
IMT connection: What is the determinant’s role in the Invertible Matrix Theorem?
Check
$A$ is invertible if and only if $\det(A) \neq 0$. A zero determinant is equivalent to all twelve IMT conditions failing simultaneously.
2x2 inverse: For $A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$ with $ad-bc \neq 0$, what is $A^{-1}$?
Check
$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$. The denominator is $\det(A)$.
If you struggled: Review The Invertible Matrix Theorem.
Try This First
Compute the determinant of $A = \begin{pmatrix}3&2\\1&4\end{pmatrix}$.
Then: find a $2 \times 2$ matrix using the same four entries ($1, 2, 3, 4$) but arranged so that the determinant equals $0$.
Show Solution
$\det\begin{pmatrix}3&2\\1&4\end{pmatrix} = 3(4) - 2(1) = 12 - 2 = 10$.
For a zero determinant using $\{1, 2, 3, 4\}$: try $\begin{pmatrix}2&4\\1&2\end{pmatrix}$: $\det = 4 - 4 = 0$.
This works because the second column is twice the first; the columns are linearly dependent, which forces the determinant to be zero (and the matrix to be non-invertible).
What the determinant measures: it is a scale factor for area. If you apply $A$ to the unit square, the image parallelogram has area $|\det(A)|$. A zero determinant means the transformation collapses the square to a line or point: the matrix “loses dimensions” and is not invertible.
The Signed Volume Idea
The determinant assigns a single number to each square matrix. Geometrically in $\mathbb{R}^2$, $|\det(A)|$ is the area of the parallelogram formed by the columns of $A$; the sign tells you whether the transformation preserves or reverses orientation. In $\mathbb{R}^3$, it is the signed volume of the parallelepiped formed by the columns.
The algebraic consequence is the most practical: the determinant is nonzero exactly when the matrix is invertible.
Prerequisite Map
Quick Reference
| Property | Value |
|---|---|
| Course | MATH 231 |
| Difficulty | Intermediate |
| Time | ~30 minutes |
| Source | Lay, Linear Algebra and Its Applications, Sections 3.1, 3.2, and 3.3 |
Key Properties at a Glance
| Property | Formula |
|---|---|
| 2x2 determinant | $\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc$ |
| Triangular matrix | $\det(A) = $ product of diagonal entries |
| Transpose | $\det(A^T) = \det(A)$ |
| Product | $\det(AB) = \det(A)\det(B)$ |
| Scalar multiple | $\det(kA) = k^n\det(A)$ for $n \times n$ |
| Invertibility | $A$ is invertible iff $\det(A) \neq 0$ |
Key Concepts
The 2x2 Determinant
For a $2 \times 2$ matrix, the determinant is defined by: \[ \det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc \]
The number $ad - bc$ is often called the “cross product” of the diagonals. If this number is zero, the columns are linearly dependent (one is a multiple of the other) and the matrix is not invertible.
Source: Lay, Linear Algebra and Its Applications, Section 3.1: Introduction to Determinants.
Cofactor Expansion
For larger matrices, the determinant is computed by cofactor expansion along any row or column.
Definitions:
- The (i,j) minor $M_{ij}$ is the $(n-1) \times (n-1)$ matrix obtained by deleting row $i$ and column $j$ from $A$.
- The (i,j) cofactor is $C_{ij} = (-1)^{i+j}\det(M_{ij})$.
The sign $(-1)^{i+j}$ follows the checkerboard pattern: \[ \begin{pmatrix}+&-&+&\cdots\\-&+&-&\cdots\\+&-&+&\cdots\\\vdots&&&\ddots\end{pmatrix} \]
Expansion along row $i$: \[ \det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + \cdots + a_{in}C_{in} \]
Expansion along column $j$: \[ \det(A) = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj} \]
The determinant is the same no matter which row or column you expand along.
Strategy: expand along the row or column with the most zeros; each zero eliminates one term.
Source: Lay, Linear Algebra and Its Applications, Section 3.1: Introduction to Determinants.
Worked Example: 3x3 Cofactor Expansion
Compute $\det\begin{pmatrix}1&3&-2\\2&1&4\\-1&0&3\end{pmatrix}$.
Expand along row 1 (no zeros, but straightforward):
\[ \det = 1 \cdot (+1)\det\begin{pmatrix}1&4\\0&3\end{pmatrix} - 3 \cdot (-1)\cdot(-1)\det\begin{pmatrix}2&4\\-1&3\end{pmatrix} + (-2) \cdot (+1)\det\begin{pmatrix}2&1\\-1&0\end{pmatrix} \]
Wait, let me be precise about signs:
- $(1,1)$ cofactor: sign $(-1)^{1+1} = +1$, minor $\det\begin{pmatrix}1&4\\0&3\end{pmatrix} = 3$
- $(1,2)$ cofactor: sign $(-1)^{1+2} = -1$, minor $\det\begin{pmatrix}2&4\\-1&3\end{pmatrix} = 6-(-4) = 10$
- $(1,3)$ cofactor: sign $(-1)^{1+3} = +1$, minor $\det\begin{pmatrix}2&1\\-1&0\end{pmatrix} = 0-(-1) = 1$
\[ \det = 1(+1)(3) + 3(-1)(10) + (-2)(+1)(1) = 3 - 30 - 2 = -29 \]
Verify by expanding along column 3 (has one entry $-2$, then $4$, then $3$):
- $(1,3)$: sign $+1$, minor $\det\begin{pmatrix}2&1\\-1&0\end{pmatrix} = 1$
- $(2,3)$: sign $-1$, minor $\det\begin{pmatrix}1&3\\-1&0\end{pmatrix} = 0-(-3) = 3$
- $(3,3)$: sign $+1$, minor $\det\begin{pmatrix}1&3\\2&1\end{pmatrix} = 1-6 = -5$
\[ \det = (-2)(+1)(1) + 4(-1)(3) + 3(+1)(-5) = -2 - 12 - 15 = -29 \]
Both expansions give $-29$. The determinant is independent of the expansion choice.
Properties of Determinants
Effect of row operations on $\det(A)$:
| Row operation | Effect on det |
|---|---|
| Swap two rows | Multiplies det by $-1$ |
| Scale a row by $k$ | Multiplies det by $k$ |
| Replace a row ($R_i \to R_i + kR_j$) | No change to det |
Triangular matrices: if $A$ is upper or lower triangular (all entries below or above the diagonal are zero), then $\det(A)$ equals the product of the diagonal entries.
Why triangular matrices are easy: cofactor expansion along the first row (or column) zeroes out all but one term at each level.
Source: Lay, Linear Algebra and Its Applications, Section 3.2: Properties of Determinants.
Determinant and Invertibility
Theorem: An $n \times n$ matrix $A$ is invertible if and only if $\det(A) \neq 0$.
This adds a 13th equivalent condition to the Invertible Matrix Theorem: $\det(A) \neq 0 \iff A$ is invertible $\iff$ all twelve prior IMT conditions hold.
Geometric meaning: $\det(A) = 0$ means the transformation $\mathbf{x} \mapsto A\mathbf{x}$ collapses at least one dimension (the columns are dependent), making it impossible to invert.
Common Pitfalls
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| $\det(A+B) = \det(A) + \det(B)$ | The determinant is multilinear in the ROWS (or columns), not in the whole matrix | Compute $\det(A+B)$ by direct expansion; there is no shortcut |
| $\det(kA) = k\,\det(A)$ | Scaling all $n$ rows by $k$ multiplies the det by $k^n$, not $k$ | For an $n \times n$ matrix, $\det(kA) = k^n\det(A)$ |
| Using the same expansion row/column for all matrices | The choice of expansion row/column is valid for any row or column | Choose the row or column with the most zeros to minimize computation |
| Forgetting the sign pattern $(-1)^{i+j}$ | Missing signs produce wrong determinant values | Apply the checkerboard sign before each cofactor; use a sign table if needed |
Why $\det(kA) = k^n\det(A)$
If $A$ is $n \times n$ and you multiply $A$ by scalar $k$ to get $kA$, then every row of $kA$ is $k$ times the corresponding row of $A$. Each of the $n$ rows contributes one factor of $k$ when you scale: pulling $k$ out of each row individually gives $k \cdot k \cdots k = k^n$ total factors.
Example ($n = 2$): $\det(2A)$ for $A = \begin{pmatrix}1&2\\3&4\end{pmatrix}$, so $2A = \begin{pmatrix}2&4\\6&8\end{pmatrix}$.
$\det(2A) = 2(8) - 4(6) = 16 - 24 = -8$.
$2^2\det(A) = 4(4-6) = 4(-2) = -8$. Confirmed.
Practice Problems
Compute the determinant of each matrix. For each, state whether the matrix is invertible.
(a) $\begin{pmatrix}5&3\\2&1\end{pmatrix}$
(b) $\begin{pmatrix}6&-2\\-3&1\end{pmatrix}$
(c) $\begin{pmatrix}4&0\\0&-7\end{pmatrix}$
Compute $\det(A)$ where $A = \begin{pmatrix}4&3&0\\2&1&-1\\0&-2&3\end{pmatrix}$.
Choose a row or column to expand along and justify your choice. Verify using a different row or column.
Use row operations (tracking their effect on the determinant) to find $\det(A)$ for:
\[ A = \begin{pmatrix}2&4&-2\\0&3&1\\-1&-2&3\end{pmatrix} \]
Let $A$ and $B$ be $3 \times 3$ matrices with $\det(A) = -3$ and $\det(B) = 5$.
Compute:
(a) $\det(AB)$
(b) $\det(A^{-1})$
(c) $\det(2B)$
(d) $\det(A^T B A)$
(a) Find the area of the parallelogram in $\mathbb{R}^2$ spanned by $\mathbf{v}_1 = \begin{pmatrix}3\\1\end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix}1\\4\end{pmatrix}$.
(b) Find the volume of the parallelepiped in $\mathbb{R}^3$ spanned by $\mathbf{u}_1 = \begin{pmatrix}2\\0\\0\end{pmatrix}$, $\mathbf{u}_2 = \begin{pmatrix}0\\3\\0\end{pmatrix}$, $\mathbf{u}_3 = \begin{pmatrix}1\\1\\4\end{pmatrix}$.
(c) If a linear transformation $T$ with matrix $A$ maps the unit square to a parallelogram of area $7$, what is $|\det(A)|$?
Common Misconceptions
$\det(A+B) = \det(A) + \det(B)$. The determinant is linear in each row separately, but it is not linear in the entire matrix. Adding two matrices and then computing the determinant is not the same as adding the determinants. A counterexample: $A = I_2$, $B = I_2$; then $A+B = 2I_2$, $\det(A+B) = 4$, but $\det(A) + \det(B) = 2$. The product formula $\det(AB) = \det(A)\det(B)$ does hold, but addition has no such shortcut.
$\det(kA) = k\,\det(A)$ for any scalar $k$. Multiplying the matrix $A$ by $k$ scales each of the $n$ rows by $k$. Each row contributes one factor of $k$ to the determinant, giving $k^n\det(A)$. The mistake arises from treating matrix scalar multiplication like multiplying one number by another. For $n=2$: $\det(kA) = k^2\det(A)$; for $n=3$: $\det(kA) = k^3\det(A)$.
cofactor expansion only works along the first row. The determinant can be expanded along any row or column, always giving the same result. Choosing the row or column with the most zero entries minimizes the number of terms to compute: each zero makes the corresponding cofactor irrelevant.
Mastery Checklist
Novice (Level 1-2):
Competent (Level 3-4):
Proficient (Level 5):
Mental Model
The Volume Machine
The determinant is the volume machine: feed in a matrix whose columns are vectors, and it outputs the signed volume of the parallelepiped those vectors span.
When the determinant is large (in absolute value), the transformation stretches space significantly. When it is between 0 and 1, the transformation compresses space. When it is exactly 0, the transformation collapses space down by at least one dimension: the matrix is singular.
The sign tells you about orientation: positive means the transformation preserves the handedness of the coordinate system; negative means it reverses it (like a reflection).
Connections
Looking ahead in MATH 231:
- Eigenvalues and Eigenvectors: eigenvalues are the solutions to $\det(A - \lambda I) = 0$; the determinant turns the eigenvalue problem into a polynomial equation
Real-world connections:
- In robotics, the Jacobian determinant tells you whether a robot arm can move independently in all directions from a given configuration; a zero Jacobian determinant signals a singular (stuck) configuration.
- In probability, the determinant of a covariance matrix of a multivariate distribution appears in the formula for the multivariate normal density.
Source: Lay, Linear Algebra and Its Applications, Section 3.1: Introduction to Determinants; Section 3.2: Properties of Determinants; Section 3.3: Cramer Rule, Volume, and Linear Transformations.
| Previous | Up | Next |
|---|---|---|
| Invertible Matrix Theorem | MATH231 Skills Index | Vector Spaces and Subspaces |
Last updated: 2026-06-16
Watch and Read
- Lynch, MATH 231 lecture playlists (YouTube)Recorded MATH 231 lectures at EWU. Use them to see each topic worked through in a full class session.