Lesson 11 of 11 · 14 min
Chapter review
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Must-know facts
17 facts
- 01Only square matrices have determinants; |A| can be negative (the bars are not modulus).
- 02det [[a, b], [c, d]] = ad − bc.
- 03Expanding along any row or column gives the same value; pick the one with most zeros.
- 04Sign of the (i, j) place is (−1)ⁱ⁺ʲ: + − + / − + − / + − +.
- 05|kA| = kⁿ|A| for A of order n; for order 3, |3A| = 27|A|.
- 06Area of a triangle = ½|det with rows (xᵢ, yᵢ, 1)|; zero area means collinear.
- 07With the area given, use both +area and −area.
- 08Mᵢⱼ: delete row i and column j. Aᵢⱼ = (−1)ⁱ⁺ʲMᵢⱼ.
- 09A row times its own cofactors gives |A|; times another row's cofactors gives 0.
- 10adj A is the transpose of the cofactor matrix; adj [[a, b], [c, d]] = [[d, −b], [−c, a]].
- 11A(adj A) = (adj A)A = |A| I.
- 12|adj A| = |A|ⁿ⁻¹; |AB| = |A||B|; |A⁻¹| = 1/|A|.
- 13A⁻¹ exists if and only if |A| ≠ 0, and A⁻¹ = (1/|A|) adj A.
- 14AX = B with |A| ≠ 0: unique solution X = A⁻¹B.
- 15|A| = 0 and (adj A)B ≠ O: no solution. |A| = 0 and (adj A)B = O: infinitely many or none.
- 16Area of the triangle (3, 8), (−4, 2), (5, 1) is 61/2 square units.
- 17Swapping two rows changes the sign; two equal or proportional rows give 0 (JEE properties).
Common traps
Where marks are lost
Formulas
10 to know
Order 2
det [[a, b], [c, d]] = ad − bc
Leading diagonal minus the other diagonal.
Order 3, along R₁
|A| = a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃
Signs + − + ; any row or column gives the same value.
Scaling
|kA| = kⁿ|A|
n is the order of A.
Area of a triangle
Area = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
Zero exactly when the three points are collinear.
Cofactor
Aᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ
Mᵢⱼ: delete row i and column j.
Adjoint identity
A(adj A) = (adj A)A = |A| I
adj A = transpose of the cofactor matrix.
Inverse
A⁻¹ = (1/|A|) adj A
Exists if and only if |A| ≠ 0.
Determinant rules
|AB| = |A||B|, |adj A| = |A|ⁿ⁻¹, |A⁻¹| = 1/|A|
A, B square of order n.
Matrix method
AX = B, |A| ≠ 0 ⇒ X = A⁻¹B
Unique solution.
Cramer's rule (JEE)
x = ∆₁/∆, y = ∆₂/∆, z = ∆₃/∆
∆ₖ: replace column k of A by B; needs ∆ ≠ 0.
Key terms
8 terms
- Determinant
- The number |A| attached to a square matrix A; ad − bc for order 2.
- Minor
- Mᵢⱼ, the determinant left after deleting the row and column of aᵢⱼ.
- Cofactor
- Aᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ, the minor with its place sign.
- Adjoint
- adj A, the transpose of the matrix of cofactors of A.
- Singular matrix
- A square matrix with determinant 0; it has no inverse.
- Non-singular matrix
- A square matrix with non-zero determinant; it is invertible.
- Consistent system
- A system of equations with at least one solution.
- Inconsistent system
- A system of equations with no solution.