Determinants Notes, Quizzes & Revision

Notes — Subtopic: Determinants

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1. What is a determinant?

A determinant is a number associated with a square matrix. It summarizes important information about the matrix such as whether it is invertible and how it scales area (2×2) or volume (3×3) under the linear transformation represented by the matrix.

2. Notation

A matrix A has determinant written as det(A) or |A|. Example: if A = [a b; c d] then det(A) = |A| = ad − bc.

3. Determinant of a 2×2 matrix (quick formula)

det(A) = ad − bc

4. Determinant of a 3×3 matrix — Sarrus rule (easy mnemonics)

For matrix

Sarrus (works only for 3×3): write first two columns again and sum products on down-right diagonals and subtract up-right diagonals:

5. Expansion by minors / cofactor expansion (general method)

To expand along row i: det(A) = Σ_j ( (−1)^(i+j) · a_ij · det(M_ij) ) where M_ij is the minor (matrix after deleting row i and column j).

Useful if a row or column has zeros — simplifies calculation.

6. Effects of elementary row/column operations on determinants

• Swapping two rows (or columns) multiplies determinant by −1.
• Multiplying a row by scalar k multiplies determinant by k. (If you multiply every row by k in an n×n matrix, determinant is multiplied by k^n.)
• Adding a multiple of one row to another row does not change the determinant.

7. Properties useful in calculations

• det(I) = 1 for identity matrix I.
• det(AB) = det(A) · det(B).
• det(A^T) = det(A).
• If A is triangular (upper or lower), det(A) = product of diagonal entries.
• Matrix A is invertible ⇔ det(A) ≠ 0. If invertible, det(A^(−1)) = 1/det(A).

8. Determinant and inverse of a 2×2 matrix

If A = [a   b
c   d] and det(A) ≠ 0, then

9. Geometric interpretation (useful for Kenya contexts: maps, land plots, engineering)

For a 2×2 matrix representing a linear map in the plane, |det(A)| is the scale factor of area. If det(A) = 0, the map collapses area onto a line (loss of dimension).

Example: If A maps the unit square to a parallelogram of area 5, then |det(A)| = 5.

In 3D, |det(A)| is the factor by which volume is scaled by the linear map A.

10. Worked example combining ideas

11. Tips for exam-style (KCSE/other Kenyan exams) calculations

• Look for rows/columns with zeros for cofactor expansion.
• Use row operations to create zeros (remember how operations affect determinant).
• For 3×3, Sarrus is quick but for larger matrices use expansion or reduce to triangular form.

12. Practice questions (try without a calculator)

1. Find det of A = [4   1
   2   3].
2. Find det of B = [1   2   3
   4   5   6
   7   8   9] (use Sarrus).
3. Given C = [2   0   1
   3   0   0
   5   1   1], compute det(C) by expansion along the second column.
4. If D is a 3×3 matrix and det(D) = 4, what is det(3D)?
5. Is matrix E = [2   4
   1   2] invertible? Explain using determinant.

13. Answers — Practice questions

1. det(A) = 4·3 − 1·2 = 12 − 2 = 10.
2. det(B) = 0 (this classic 3×3 with arithmetic progression rows is singular).
3. Expanding C along column 2 (entries 0,0,1): only third row contributes → det(C) = (+1)·cofactor = 1·det([2 1; 3 0])·(−1)^(3+2) = (−1)·(2·0 − 1·3) = (−1)·(−3) = 3. (Alternatively expand along row/col giving det=3.)
4. det(3D) = 3^3 · det(D) = 27 · 4 = 108 (multiply by k^n for an n×n matrix).
5. det(E) = 2·2 − 4·1 = 4 − 4 = 0, so E is not invertible (singular).

These notes are written to fit Kenyan secondary-level contexts (exams and practical uses). For more practice, try finding determinants after performing elementary row operations—this builds speed for exams.