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Notes / Unit 4 · Module 3 · §3.1

Determinants

Every square matrix has a number attached to it, its determinant. The exam asks you to find a 2 × 2 determinant, the minors and cofactors of a 3 × 3 or 4 × 4, a determinant by cofactor expansion along the row or column with the most zeros, and a triangular determinant by multiplying the main diagonal.

4.1The determinant of a 2 × 2 matrix

Every square matrix A has a real number called its determinant, written det(A) or |A|. Determinants first came out of patterns in the solutions of linear systems: when the coefficient matrix has a nonzero determinant, the system has exactly one solution.

det(A) = |A| = a₁₁a₂₂ − a₁₂a₂₁

aᵢⱼ
the entry in row i, column j of the 2 × 2 matrix A

The product of the main diagonal minus the product of the other diagonal. The result can be positive, zero or negative.

A = [ 2  −3 ]
[ 1   2 ]
Example 1a

Worked example

Example 1

  1. 1a) |A| for the matrix above = (2)(2) − (−3)(1) = 4 + 3 = 7.
  2. 2b) B = [2 1; 4 2]: |B| = (2)(2) − (1)(4) = 4 − 4 = 0.
  3. 3c) C = [0 3/2; 2 4]: |C| = (0)(4) − (3/2)(2) = 0 − 3 = −3.

Worked example

Exercise 1

  1. 1a) [1 2; 3 4]: (1)(4) − (2)(3) = 4 − 6 = −2.
  2. 2b) [−1 0; 2 −3]: (−1)(−3) − (0)(2) = 3.
  3. 3c) [4 −5; −6 7]: (4)(7) − (−5)(−6) = 28 − 30 = −2.

4.2Minors and cofactors

Beyond 2 × 2, a determinant is built out of smaller ones, through minors and cofactors.

Minor Mᵢⱼ
The determinant of the matrix left after deleting row i and column j of A. A number, with no sign attached.
Cofactor Cᵢⱼ
Cᵢⱼ = (−1)^(i+j) Mᵢⱼ. The minor, with its sign flipped when i + j is odd.

The signs (−1)^(i+j) form a checkerboard that starts with + in the top-left corner. Read the sign of Cᵢⱼ off it instead of computing the power.

3 × 3        4 × 4
[ +  −  + ]  [ +  −  +  − ]
[ −  +  − ]  [ −  +  −  + ]
[ +  −  + ]  [ +  −  +  − ]
              [ −  +  −  + ]
The sign of Cᵢⱼ relative to Mᵢⱼ

Worked example

For a 3 × 3 matrix, the entries a₂₁ and a₂₂

  1. 1M₂₁: delete row 2 and column 1, then take the 2 × 2 determinant of what is left: M₂₁ = a₁₂a₃₃ − a₁₃a₃₂.
  2. 2C₂₁ = (−1)^(2+1) M₂₁ = −M₂₁, since 3 is odd.
  3. 3M₂₂: delete row 2 and column 2: M₂₂ = a₁₁a₃₃ − a₁₃a₃₁.
  4. 4C₂₂ = (−1)^(2+2) M₂₂ = M₂₂, since 4 is even.

4.3Every minor and cofactor of a 3 × 3

A = [ 0   2  1 ]
    [ 3  −1  2 ]
    [ 4   0  1 ]
Example 2

Worked example

Example 2: the minors, then the cofactors

  1. 1Row 1: M₁₁ = (−1)(1) − (2)(0) = −1, M₁₂ = (3)(1) − (2)(4) = −5, M₁₃ = (3)(0) − (−1)(4) = 4.
  2. 2Row 2: M₂₁ = (2)(1) − (1)(0) = 2, M₂₂ = (0)(1) − (1)(4) = −4, M₂₃ = (0)(0) − (2)(4) = −8.
  3. 3Row 3: M₃₁ = (2)(2) − (1)(−1) = 5, M₃₂ = (0)(2) − (1)(3) = −3, M₃₃ = (0)(−1) − (2)(3) = −6.
  4. 4Apply the checkerboard + − + / − + − / + − +: C₁₁ = −1, C₁₂ = 5, C₁₃ = 4.
  5. 5C₂₁ = −2, C₂₂ = −4, C₂₃ = 8.
  6. 6C₃₁ = 5, C₃₂ = 3, C₃₃ = −6.

4.4The determinant of a square matrix

For a square matrix of order n ≥ 2, the determinant is the sum of the entries in the first row, each multiplied by its own cofactor. This is called expanding by cofactors in the first row.

|A| = a₁₁C₁₁ + a₁₂C₁₂ + ⋯ + a₁ₙC₁ₙ

For a 3 × 3, written with minors: |A| = a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃.

Worked example

Example 3: |A| for the matrix of Example 2

  1. 1Row 1 is 0, 2, 1 and its cofactors are C₁₁ = −1, C₁₂ = 5, C₁₃ = 4.
  2. 2|A| = (0)(−1) + (2)(5) + (1)(4) = 0 + 10 + 4 = 14.

4.5Expansion along any row or column

|A| = aᵢ₁Cᵢ₁ + aᵢ₂Cᵢ₂ + ⋯ + aᵢₙCᵢₙ (row i) |A| = a₁ⱼC₁ⱼ + a₂ⱼC₂ⱼ + ⋯ + aₙⱼCₙⱼ (column j)

Theorem 3.1, Expansion by Cofactors: every row and every column gives the same determinant.

So choose the line that is least work: the row or column with the most zeros. A zero entry kills its term aᵢⱼCᵢⱼ, so its cofactor never has to be found.

Worked example

Exercise 2: Example 2's matrix again, along other lines

  1. 1Column 2 is 2, −1, 0 with signs − + −, so |A| = −(2)(−5) + (−1)(−4) − (0)(−3) = 10 + 4 + 0 = 14.
  2. 2Row 3 is 4, 0, 1 with signs + − +, so |A| = (4)(5) − (0)(−3) + (1)(−6) = 20 + 0 − 6 = 14.
  3. 3Both agree with the first-row expansion of Example 3. Row 3 and column 2 each have a zero, so each needs only two 2 × 2 determinants.

4.6A 4 × 4 by cofactor expansion

A 4 × 4 expansion has four terms, each needing a 3 × 3 determinant. Look for a row or column that is mostly zeros first; it can cut the work to a single 3 × 3.

A = [  1  −2  3   0 ]
    [ −1   1  0   2 ]
    [  0   2  0   3 ]
    [  3   4  0  −2 ]
Exercise 3

Worked example

Exercise 3: expand along column 3

  1. 1Column 3 is 3, 0, 0, 0: three zeros, so |A| = a₁₃C₁₃ = 3C₁₃.
  2. 21 + 3 = 4 is even, so C₁₃ = M₁₃: delete row 1 and column 3, leaving [−1 1 2; 0 2 3; 3 4 −2].
  3. 3Expand that along its column 1, which has a zero: M₁₃ = (−1)[(2)(−2) − (3)(4)] − 0 + (3)[(1)(3) − (2)(2)] = (−1)(−16) + (3)(−1) = 16 − 3 = 13.
  4. 4|A| = 3 · 13 = 39.

4.7Triangular and diagonal matrices

Upper triangular
Every entry below the main diagonal is 0.
Lower triangular
Every entry above the main diagonal is 0.
Diagonal
Both upper and lower triangular: every entry off the main diagonal is 0.

|A| = a₁₁a₂₂a₃₃ ⋯ aₙₙ

Theorem 3.2: the determinant of a triangular matrix of order n is the product of its main diagonal. Expanding along the column or row of zeros each time leaves exactly that product.

A = [  2   0  0  0 ]
    [  4  −2  0  0 ]
    [ −5   6  1  0 ]
    [  1   5  3  3 ]
Example 4, lower triangular

Worked example

Example 4

  1. 1Every entry above the main diagonal is 0, so A is lower triangular.
  2. 2|A| = (2)(−2)(1)(3) = −12. The entries below the diagonal play no part.

A diagonal matrix is triangular too, so the same rule holds: the identity Iₙ has |Iₙ| = 1, and 3I₃ has determinant 3 · 3 · 3 = 27.

Check yourself

Answer these out loud, without scrolling up. Any you can't, reread that section — then let the drill mark you.

  1. 1.Find |A| for A = [3 −2; 4 1], and say which product is subtracted from which.
  2. 2.What is the difference between the minor M₃₂ and the cofactor C₃₂, and which rows and columns do you delete to get them?
  3. 3.Write the checkerboard of signs for a 4 × 4. Is C₂₄ equal to M₂₄ or to −M₂₄?
  4. 4.Theorem 3.1 says you may expand along any row or column. Which one do you choose, and why?
  5. 5.Find the determinant of the 4 × 4 in Exercise 3 without looking, by expanding along its column of zeros.
  6. 6.Why is the determinant of a triangular matrix the product of its main diagonal, and when is it 0?