Lesson 1 of 11 · 7 min
What a determinant is
NCERT §4.1–4.2.1
The school stationery counter has lost its price list. Two old bills survive: 3 pens and 2 exercise books for ₹95, and 1 pen and 4 books for ₹165. Can the prices be rebuilt from them?
The story this chapter follows: The counter's lost price list
The lesson in notes
In short
Every square matrix A = [aᵢⱼ] of order n has a number attached to it, its determinant, written |A|, det A or ∆. It can be read as a function from square matrices to numbers.
|A| is read 'determinant of A'. The bars do not mean modulus: a determinant can be negative, as det [[2, 4], [5, 1]] = 2 − 20 = −18 shows.
Only square matrices have determinants. A 2 × 3 matrix has none.
For a 1 × 1 matrix [a], the determinant is the entry itself: |[a]| = a, even when a is negative.
Where it comes from: the pair a₁x + b₁y = c₁, a₂x + b₂y = c₂ has exactly one solution when a₁b₂ − a₂b₁ ≠ 0. That number, formed from the coefficient matrix, is its determinant, and it 'determines' whether the solution is unique.
This chapter works with determinants of order up to three with real entries, and uses them for areas, inverses and systems of linear equations.