Lesson 11 of 11 · 14 min

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Must-know facts

17 facts

1Only square matrices have determinants; |A| can be negative (the bars are not modulus).
2det [[a, b], [c, d]] = ad − bc.
3Expanding along any row or column gives the same value; pick the one with most zeros.
4Sign of the (i, j) place is (−1)ⁱ⁺ʲ: + − + / − + − / + − +.
5|kA| = kⁿ|A| for A of order n; for order 3, |3A| = 27|A|.
6Area of a triangle = ½|det with rows (xᵢ, yᵢ, 1)|; zero area means collinear.
7With the area given, use both +area and −area.
8Mᵢⱼ: delete row i and column j. Aᵢⱼ = (−1)ⁱ⁺ʲMᵢⱼ.
9A 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

Reading |A| as a modulus and writing it as positive.

A determinant is a signed number: det [[2, 4], [5, 1]] = −18. Only the area formula takes an absolute value.

Writing |2A| = 2|A| for a 3 × 3 matrix.

Each of the n rows gets a factor 2: |2A| = 2³|A| = 8|A|.

Forgetting the (−1)ⁱ⁺ʲ sign when turning a minor into a cofactor.

Check the chessboard: positions with i + j odd, such as (1, 2) and (2, 1), flip the sign.

Using the cofactor matrix itself as adj A.

Transpose it: the cofactors of row i become column i of adj A.

Taking only +area when a coordinate is found from a given area.

Solve det-expression = ±2 × area; both signs usually give answers, as k = ±2 in the D(k, 0) example.

Calling every system with |A| = 0 inconsistent.

Compute (adj A)B. Only (adj A)B ≠ O proves there is no solution; (adj A)B = O leaves both possibilities open.

Writing X = BA⁻¹ for AX = B.

Multiply on the left by A⁻¹: X = A⁻¹B.

Writing |adj A| = |A|ⁿ.

It is |A|ⁿ⁻¹, from |adj A||A| = |A|ⁿ; for order 3, |adj A| = |A|².

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.