Determinants: Computation, Properties & Cofactors

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# Determinants: Computation, Properties & Cofactors ## 🎯 Learning Objectives After this topic you will be able to: - Compute the determinant of $2 \times 2$ and $3 \times 3$ matrices - Use Sarrus's rule for $3 \times 3$ determinants - Compute determinants by cofactor expansion along any row or column - List and app...

Determinants: Computation, Properties & Cofactors

🎯 Learning Objectives

After this topic you will be able to:

Compute the determinant of $2 \times 2$ and $3 \times 3$ matrices
Use Sarrus's rule for $3 \times 3$ determinants
Compute determinants by cofactor expansion along any row or column
List and apply the key properties of determinants
Use the determinant to test whether a matrix is invertible
Compute minors and cofactors

📋 Prerequisites

Matrices Introduction (this week) — you need to be comfortable with matrix notation and dimensions
Basic algebra — arithmetic, factorisation
Determinants are a bridge between matrices (Week 1) and solving linear systems (Week 2)

1. What is a Determinant?

1.1 Intuition: The "Scaling Factor" of a Matrix

A determinant answers the question: by what factor does this matrix scale areas (or volumes)? Imagine a

2 \times 2

matrix as a transformation that maps the unit square to a parallelogram. The absolute value of the determinant is the area of that parallelogram. If the determinant is zero, the parallelogram collapses — the transformation squishes everything into a lower dimension. (Diagram) A zero determinant means the matrix is singular (non-invertible) — it destroys information. A non-zero determinant means the matrix is invertible — the transformation can be reversed.

1.2 Formal Definition

Definition (Determinant). The determinant is a function $\det : \mathbb{R}^{n \times n} \to \mathbb{R}$ that assigns to each square matrix $A$ a real number $\det(A)$ , satisfying:

$\det(I_n) = 1$

$\det(AB) = \det(A)\det(B)$

$\det$ is alternating multilinear in the columns (or rows) We'll compute it concretely below before worrying about the abstract properties.

2. Computing Determinants

2.1 $2 \times 2$ Determinant

For

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

, the determinant is:

> \det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc >

Example 1: 2×2 determinant

> \det\begin{pmatrix} 3 & 1 \\ 4 & -2 \end{pmatrix} = 3(-2) - 1(4) = -6 - 4 = -10 >

2.2 $3 \times 3$ Determinant — Sarrus's Rule

For

3 \times 3

matrices, Sarrus's rule is a handy shortcut (but only works for

3 \times 3

!).

For

> A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} >

:

Copy the first two columns to the right

Sum of products along the three downward diagonals (blue)

Minus sum of products along the three upward diagonals (red)

> \det(A) = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} >

Example 2: Sarrus's rule

> A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} >

Copy columns and compute:

> \begin{matrix} 1 & 2 & 3 & | & 1 & 2 \\ 4 & 5 & 6 & | & 4 & 5 \\ 7 & 8 & 9 & | & 7 & 8 \end{matrix} >

> \begin{aligned} \det(A) &= 1\cdot5\cdot9 + 2\cdot6\cdot7 + 3\cdot4\cdot8 \\ &\quad - 3\cdot5\cdot7 - 1\cdot6\cdot8 - 2\cdot4\cdot9 \\ &= 45 + 84 + 96 - 105 - 48 - 72 \\ &= 0 \end{aligned} >

This matrix is singular (determinant = 0). Example 3: Sarrus with a non-zero determinant

> A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 0 & 2 & -1 \end{bmatrix} >

> \begin{aligned} \det(A) &= 1(3)(-1) + 0(1)(0) + 2(-1)(2) \\ &\quad - 2(3)(0) - 1(1)(2) - 0(-1)(-1) \\ &= -3 + 0 - 4 - 0 - 2 - 0 \\ &= -9 \end{aligned} >

2.3 Cofactor Expansion (General Method)

For larger matrices (

n \geq 4

), Sarrus doesn't work. We use cofactor expansion (also called Laplace expansion).

Definition (Minor and Cofactor).

Minor $M_{ij}$ : the determinant of the $(n-1) \times (n-1)$ matrix obtained by deleting row $i$ and column $j$ from $A$ .

Cofactor $C_{ij} = (-1)^{i+j} M_{ij}$ . Cofactor Expansion Theorem. For any square matrix $A$ ,

> \det(A) = \sum_{j=1}^{n} a_{ij} C_{ij} \quad \text{(expand along row $i$)} >

or equivalently along any column $j$ :

> \det(A) = \sum_{i=1}^{n} a_{ij} C_{ij} >

The sign pattern

(-1)^{i+j}

is a checkerboard:

\begin{bmatrix} + & - & + & - & \dots \\ - & + & - & + & \dots \\ + & - & + & - & \dots \\ - & + & - & + & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}

Example 4: Cofactor expansion along row 1

Compute $\det(A)$ for

> A = \begin{bmatrix} 2 & 1 & 0 \\ -3 & 2 & 1 \\ 1 & 0 & -2 \end{bmatrix} >

.

Step 1: Identify $a_{11} = 2$ , $a_{12} = 1$ , $a_{13} = 0$ .

Step 2: Compute minors and cofactors.

> M_{11} = \det\begin{bmatrix} 2 & 1 \\ 0 & -2 \end{bmatrix} = 2(-2) - 1(0) = -4 >

, $C_{11} = (+1)(-4) = -4$

> M_{12} = \det\begin{bmatrix} -3 & 1 \\ 1 & -2 \end{bmatrix} = (-3)(-2) - 1(1) = 6 - 1 = 5 >

, $C_{12} = (-1)(5) = -5$

> M_{13} = \det\begin{bmatrix} -3 & 2 \\ 1 & 0 \end{bmatrix} = (-3)(0) - 2(1) = -2 >

, $C_{13} = (+1)(-2) = -2$

Step 3: $\det(A) = 2(-4) + 1(-5) + 0(-2) = -8 - 5 + 0 = -13$

Check: Use Sarrus to verify: $2(2)(-2) + 1(1)(1) + 0(-3)(0) - 0(2)(1) - 2(1)(0) - 1(-3)(-2)$ $= -8 + 1 + 0 - 0 - 0 - 6 = -13$ ✓ Example 5: Smart expansion — choose the row with the most zeros

Compute $\det(A)$ for

> A = \begin{bmatrix} 3 & 0 & 2 & 0 \\ 1 & 2 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 2 & 1 & 1 & 3 \end{bmatrix} >

.

Expand along row 3 (has three zeros, only one non-zero entry at column 3):

$a_{31} = 0$ , $a_{32} = 0$ , $a_{33} = -1$ , $a_{34} = 0$ .

$\det(A) = (-1) \cdot C_{33}$ where $C_{33} = (+1) \cdot M_{33}$ (since $3+3=6$ , even).

> M_{33} = \det\begin{bmatrix} 3 & 0 & 0 \\ 1 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix} >

(delete row 3, column 3)

Expand again along row 1:

> M_{33} = 3 \cdot \det\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} - 0 + 0 = 3(6 - 1) = 15 >

Therefore $\det(A) = -1 \cdot 15 = -15$ .

3. Properties of Determinants

These properties are essential for simplifying computations and for proofs.

Property	Statement	Example
P1: Identity	$\det(I_n) = 1$
P2: Row swap	Swapping two rows changes sign	$\det\begin{vmatrix} c & d \\ a & b \end{vmatrix} = -(ad - bc)$
P3: Scalar multiple of a row	If a row is multiplied by $c$ , $\det$ multiplies by $c$	$\det\begin{vmatrix} ca & cb \\ c & d \end{vmatrix} = c(ad - bc)$
P4: Add multiple of one row to another	$\det$ unchanged	This is why row reduction preserves determinants
P5: Zero row/column	$\det(A) = 0$ if any row or column is all zeros
P6: Identical rows	$\det(A) = 0$ if two rows are identical
P7: Triangular	$\det(A) = \prod a_{ii}$ for triangular matrices	Product of diagonal entries
P8: Product	$\det(AB) = \det(A)\det(B)$
P9: Transpose	$\det(A^T) = \det(A)$
P10: Inverse	$\det(A^{-1}) = 1/\det(A)$	Only when $\det(A) \neq 0$
P11: Row linearity	$\det$ is linear in each row	$\det(\text{row}_1 + \text{row}'_1, \dots) = \det(\text{row}_1, \dots) + \det(\text{row}'_1, \dots)$

Example 6: Using properties to simplify

Compute

> \det\begin{vmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 5 & 7 \end{vmatrix} >

.

Notice row 2 is twice row 1: $2 \times (1, 2, 3) = (2, 4, 6)$ .

By P6 (or by P3 + P11), since rows 1 and 2 are linearly dependent, the determinant is $0$ .

4. Determinant and Invertibility

Theorem (Invertibility Criterion). An $n \times n$ matrix $A$ is invertible if and only if $\det(A) \neq 0$ . This is arguably the most important property of determinants. It tells us:

$\det(A) \neq 0$ $\Longleftrightarrow$ $A$ is invertible $\Longleftrightarrow$ $A\mathbf{x} = \mathbf{b}$ has a unique solution for every $\mathbf{b}$
$\det(A) = 0$ $\Longleftrightarrow$ $A$ is singular $\Longleftrightarrow$ $A\mathbf{x} = \mathbf{b}$ has either no solution or infinitely many

Proof sketch: Why det ≠ 0 ⇔ invertible

$A$ is invertible iff its RREF is $I_n$ . Row operations only multiply the determinant by non-zero scalars (or flip sign). So $\det(A) \neq 0$ iff $\det(\text{RREF}(A)) \neq 0$ . But the only $n \times n$ matrix in RREF with non-zero determinant is $I_n$ , whose determinant is 1. Therefore $\det(A) \neq 0$ iff RREF $(A) = I_n$ iff $A$ is invertible.

5. Minors and Cofactors — Formal Definitions

Definition (Minor). Given $A_{n \times n}$ , the minor $M_{ij}$ is the determinant of the submatrix formed by deleting row $i$ and column $j$ . Definition (Cofactor). $C_{ij} = (-1)^{i+j} M_{ij}$ . The cofactor matrix (matrix of cofactors) and its transpose — the adjugate (or classical adjoint) — will appear in Week 2 when we compute inverses and use Cramer's rule. Example 7: Finding all minors and cofactors for a 3×3 matrix

> A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & -1 & 4 \\ 2 & 1 & -2 \end{bmatrix} >

> M_{11} = \det\begin{bmatrix} -1 & 4 \\ 1 & -2 \end{bmatrix} = (-1)(-2) - 4(1) = 2 - 4 = -2 >

, $C_{11} = (+1)(-2) = -2$

> M_{12} = \det\begin{bmatrix} 0 & 4 \\ 2 & -2 \end{bmatrix} = 0(-2) - 4(2) = -8 >

, $C_{12} = (-1)(-8) = 8$

> M_{13} = \det\begin{bmatrix} 0 & -1 \\ 2 & 1 \end{bmatrix} = 0(1) - (-1)(2) = 2 >

, $C_{13} = (+1)(2) = 2$

> M_{21} = \det\begin{bmatrix} 2 & 3 \\ 1 & -2 \end{bmatrix} = 2(-2) - 3(1) = -4 - 3 = -7 >

, $C_{21} = (-1)(-7) = 7$

... and so on.

6. Edge Cases & Gotchas

Situation	What Happens
$1 \times 1$ matrix	$\det([a]) = a$
$0 \times 0$ matrix	Convention: $\det = 1$ (vacuously)
Row swap	Flips the sign. A common exam trick: do an odd number of swaps
Determinant of $cA$	$\det(cA) = c^n \det(A)$ , not $c\det(A)$ — because every row gets multiplied by $c$
$\det(A+B)$	$\det(A+B) \neq \det(A) + \det(B)$ in general. This is a common trap

7. Common Pitfalls

Students often forget

(-1)^{i+j}

. Remember the checkerboard: top-left is

+

❌ Pitfall 2: Applying Sarrus to $4 \times 4$ or larger

Sarrus only works for

3 \times 3

. For larger matrices, use cofactor expansion or row reduction.

❌ Pitfall 3: Thinking $\det(A+B) = \det(A) + \det(B)$

Determinants are not linear in addition; they are linear in each row.

\det(A+B)

is almost never equal to

\det(A) + \det(B)

8. Formula Summary Table

Concept	Formula	Notes
2×2 determinant	$ad - bc$
3×3 determinant (Sarrus)	$a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}$	Only for $3 \times 3$
Cofactor	$C_{ij} = (-1)^{i+j} M_{ij}$	$M_{ij}$ = minor
Cofactor expansion	$\det(A) = \sum_j a_{ij} C_{ij}$	Along any row or column
Triangular determinant	$\prod_i a_{ii}$	Product of diagonal entries
Invertibility test	$\det(A) \neq 0$ $\iff$ $A$ is invertible

9. 📝 Practice Questions

Q1: 2×2 determinant

Compute

> \det\begin{pmatrix} 5 & -2 \\ 3 & 4 \end{pmatrix} >

.

Strategy: $ad - bc$ .

Solution: $\det = 5(4) - (-2)(3) = 20 + 6 = 26$ . Q2: 3×3 determinant using Sarrus

> \det\begin{pmatrix} 1 & 0 & -1 \\ 2 & 1 & 3 \\ 0 & -2 & 1 \end{pmatrix} >

Strategy: Use Sarrus or cofactor expansion.

Solution (Sarrus):

> \begin{aligned} \det &= 1(1)(1) + 0(3)(0) + (-1)(2)(-2) \\ &\quad - (-1)(1)(0) - 1(3)(-2) - 0(2)(1) \\ &= 1 + 0 + 4 - 0 + 6 - 0 = 11 \end{aligned} >

Q3: Determinant using row operations

Use row operations to simplify then compute

> \det\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{pmatrix} >

.

Strategy: Subtract 4×row1 from row2, 7×row1 from row3. Determinant unchanged by these operations.

Solution:

> \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{vmatrix} = \begin{vmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & -6 & -11 \end{vmatrix} >

Now expand along column 1:

> = 1 \cdot \det\begin{pmatrix} -3 & -6 \\ -6 & -11 \end{pmatrix} = (-3)(-11) - (-6)(-6) = 33 - 36 = -3 >

Q4: Triangular determinant

Compute

> \det\begin{pmatrix} 2 & 0 & 0 & 0 \\ 3 & -1 & 0 & 0 \\ 4 & 5 & 3 & 0 \\ 1 & -2 & 0 & 4 \end{pmatrix} >

.

Strategy: For a lower triangular matrix, multiply diagonal entries.

Solution: $\det = 2(-1)(3)(4) = -24$ . Q5: Cofactor expansion

Compute $\det(A)$ by expanding along column 2:

> A = \begin{pmatrix} 1 & 0 & 2 \\ 3 & 4 & 5 \\ 0 & -1 & 2 \end{pmatrix} >

.

Strategy: $\det = \sum_i a_{i2} C_{i2}$ .

Solution: $a_{12} = 0$ , $a_{22} = 4$ , $a_{32} = -1$ .

> C_{12} = (-1)^{1+2} M_{12} = -\det\begin{pmatrix} 3 & 5 \\ 0 & 2 \end{pmatrix} = -(6 - 0) = -6 >

> C_{22} = (+1) M_{22} = \det\begin{pmatrix} 1 & 2 \\ 0 & 2 \end{pmatrix} = 2 - 0 = 2 >

> C_{32} = (-1)^{3+2} M_{32} = -\det\begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix} = -(5 - 6) = 1 >

$\det = 0(-6) + 4(2) + (-1)(1) = 0 + 8 - 1 = 7$ Q6: Determinant of a product

If $\det(A) = 3$ and $\det(B) = -2$ , find $\det(AB)$ and $\det(2A)$ for $A, B$ $3 \times 3$ .

Strategy: Use $\det(AB) = \det(A)\det(B)$ and $\det(cA) = c^n \det(A)$ .

Solution: $\det(AB) = 3(-2) = -6$ . $\det(2A) = 2^3 \cdot 3 = 8 \cdot 3 = 24$ . Q7: Row swap effect

If $\det(A) = 10$ and $B$ is obtained by swapping rows 1 and 3 of $A$ , what is $\det(B)$ ?

Solution: Swapping two rows changes the sign. $\det(B) = -10$ . Q8: Determinant of inverse

If $A$ is $4 \times 4$ and $\det(A) = -5$ , find $\det(A^{-1})$ .

Solution: $\det(A^{-1}) = 1/\det(A) = -1/5$ . Q9: Singular matrix

Find $k$ so that

> A = \begin{pmatrix} k & 2 \\ 3 & k-1 \end{pmatrix} >

is singular.

Strategy: Set $\det(A) = 0$ .

Solution: $\det(A) = k(k-1) - 6 = k^2 - k - 6 = (k-3)(k+2) = 0$ . So $k = 3$ or $k = -2$ . Q10: Volume via determinant

Find the volume of the parallelepiped spanned by $\mathbf{a} = (1,0,2)$ , $\mathbf{b} = (0,1,1)$ , $\mathbf{c} = (2,1,0)$ .

Strategy: Volume = $|\det([\mathbf{a}\; \mathbf{b}\; \mathbf{c}])|$ .

Solution:

> \det\begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 2 & 1 & 0 \end{pmatrix} = 1(1\cdot0 - 1\cdot1) - 0 + 2(0\cdot1 - 1\cdot2) = 1(-1) + 2(-2) = -1 - 4 = -5 >

Volume = $|-5| = 5$ .

🔗 Cross-References

Next topic: Gaussian Elimination — using row operations and determinants to solve systems
Week 2 (Cramer's Rule): Expresses solutions using determinants
Week 5 (Rank-Nullity): Determinant zero ⇒ rank deficient
Week 11 (Hessian): The Hessian's determinant determines the nature of critical points
BSMA1001 (Maths 1): The area/volume interpretation connects to integrals Join Discord Previous1.2 Matrices IntroductionNext2.1 Gaussian Elimination

What this note is really saying

Determinants: Computation, Properties & Cofactors

🎯 Learning Objectives

📋 Prerequisites

1. What is a Determinant?

1.1 Intuition: The "Scaling Factor" of a Matrix

1.2 Formal Definition

2. Computing Determinants

2.1 2×22 \times 22×2 Determinant

2.2 3×33 \times 33×3 Determinant — Sarrus's Rule

2.3 Cofactor Expansion (General Method)

3. Properties of Determinants

4. Determinant and Invertibility

5. Minors and Cofactors — Formal Definitions

6. Edge Cases & Gotchas

7. Common Pitfalls

❌ Pitfall 1: Forgetting the alternating sign in cofactor expansion

❌ Pitfall 2: Applying Sarrus to 4×44 \times 44×4 or larger

❌ Pitfall 3: Thinking det⁡(A+B)=det⁡(A)+det⁡(B)\det(A+B) = \det(A) + \det(B)det(A+B)=det(A)+det(B)

8. Formula Summary Table

9. 📝 Practice Questions

🔗 Cross-References

Intelligence Hub

2.1 $2 \times 2$ Determinant

2.2 $3 \times 3$ Determinant — Sarrus's Rule

❌ Pitfall 2: Applying Sarrus to $4 \times 4$ or larger

❌ Pitfall 3: Thinking $\det(A+B) = \det(A) + \det(B)$