Determinants
The scalar companion of square matrices β solving equations, finding areas, and testing invertibility
In this chapter
4.1 What is a Determinant?
"All Mathematical truths are relative and conditional." β C.P. Steinmetz
A determinant is a scalar value associated with every square matrix. It encodes important properties of the matrix β whether it's invertible, how it transforms area/volume, and whether a system of linear equations has a unique solution.
For a square matrix A = [aα΅’β±Ό] of order n, the determinant is denoted by |A| or det(A) or β.
Order 1: |[a]| = a
Order 2: |a b; c d| = ad β bc
Order 3: Expand along any row or column
Determinant of Order 3 Γ 3
A 3Γ3 determinant can be expanded along any row or column (6 ways total), and all give the same result. Expanding along the first row:
Always expand along the row or column with the most zeros β it saves time! The sign pattern for positions alternates: +, β, +; β, +, β; +, β, +
If A = kB where A and B are square matrices of order n, then |A| = kβΏ|B|, not k|B|.
Example: If A = 2B (2Γ2), then |A| = 4|B|, not 2|B|.
4.2 Area of a Triangle
Determinants provide an elegant way to compute the area of a triangle when the vertex coordinates are known.
xβ yβ 1
xβ yβ 1
xβ yβ 1
(i) Area is always positive β take the absolute value of the determinant
(ii) If the area is given, use both Β± values for the determinant
(iii) If three points are collinear, the area = 0
4.3 Minors and Cofactors
Minors and cofactors provide a systematic way to expand determinants and are essential for finding adjoints and inverses.
Minor (Mα΅’β±Ό)
The determinant obtained by deleting the ith row and jth column. For a 3Γ3 matrix, each minor is a 2Γ2 determinant.
Cofactor (Aα΅’β±Ό)
Aα΅’β±Ό = (β1)β±βΊΚ² Γ Mα΅’β±Ό. The cofactor is the minor with a sign depending on position.
Determinant via Cofactor Expansion
= aββAββ + aββAββ + aββAββ (expanding along Cβ)
The sum of products of elements of any row (or column) with their own cofactors = determinant.
But if elements of one row are multiplied with cofactors of a different row, the sum = 0.
4.4 Adjoint and Inverse
The adjoint of a matrix is the transpose of its cofactor matrix. It is the key to finding the inverse.
Adjoint of a Matrix
The adjoint of a square matrix A is the transpose of the matrix of cofactors:
adj A = [Aβ±Όα΅’] (transpose of cofactor matrix [Aα΅’β±Ό])
Shortcut for 2Γ2: adj [a, b; c, d] = [d, βb; βc, a] β swap diagonal, negate off-diagonal
Key Theorem
Inverse of a Matrix
Non-singular (|A| β 0)
Aβ»ΒΉ exists and is unique. Aβ»ΒΉ = (1/|A|) adj A
Singular (|A| = 0)
Aβ»ΒΉ does NOT exist. The matrix cannot be inverted.
Properties
| Property | Statement |
|---|---|
| Inverse Formula | Aβ»ΒΉ = (1/|A|) adj A |
| |adj A| | |adj A| = |A|βΏβ»ΒΉ (n = order) |
| Reversal Law | (AB)β»ΒΉ = Bβ»ΒΉAβ»ΒΉ |
| Double Inverse | (Aβ»ΒΉ)β»ΒΉ = A |
| Product Rule | |AB| = |A| Γ |B| |
4.5 Applications
Determinants and matrix inverses are powerful tools for solving systems of linear equations and checking their consistency.
Solving AX = B
For the system aβx + bβy + cβz = dβ, etc., write AX = B:
Case 1: |A| β 0 β Unique solution: X = Aβ»ΒΉB
Case 2: |A| = 0 and (adj A)B β O β No solution (inconsistent)
Case 3: |A| = 0 and (adj A)B = O β Either no solution or infinitely many
Consistency
Consistent
The system has at least one solution (unique or infinite).
Inconsistent
The system has no solution at all.
4.6 Summary
β’ Determinant is a scalar associated with square matrices β |A| or det(A)
β’ 2Γ2: |a b; c d| = ad β bc Β· 3Γ3: expand along any row/column
β’ |kA| = kβΏ|A| (n = order), not k|A|
β’ Area of triangle = Β½|determinant with vertex coordinates|
β’ Minor Mα΅’β±Ό: delete row i, column j Β· Cofactor Aα΅’β±Ό = (β1)β±βΊΚ² Mα΅’β±Ό
β’ adj A = transpose of cofactor matrix Β· A(adj A) = |A|I
β’ Singular: |A| = 0 (no inverse) Β· Non-singular: |A| β 0 (inverse exists)
β’ Aβ»ΒΉ = (1/|A|) adj A when |A| β 0
β’ |adj A| = |A|βΏβ»ΒΉ Β· |AB| = |A||B| Β· (AB)β»ΒΉ = Bβ»ΒΉAβ»ΒΉ
β’ Solve AX = B β X = Aβ»ΒΉB when |A| β 0