Quick Summary
Matrices provide a compact way to handle systems of equations and linear transformations. Determinants are scalars derived from square matrices that encode important properties. Together they form the foundation for solving simultaneous equations, 3D geometry, and transformations in higher classes.
Types of Matrices
| Type | Definition |
|---|---|
| Row matrix | 1 × n matrix |
| Column matrix | m × 1 matrix |
| Square matrix | m × m matrix |
| Diagonal matrix | Square; non-diagonal elements = 0 |
| Identity matrix (I) | Diagonal matrix; all diagonal elements = 1 |
| Zero matrix (O) | All elements = 0 |
| Symmetric | Aᵀ = A |
| Skew-symmetric | Aᵀ = −A; diagonal elements = 0 |
| Upper/Lower triangular | Elements below/above diagonal = 0 |
Matrix Operations
Addition/Subtraction: Only for matrices of the same order. Done element-wise.
Scalar multiplication: kA multiplies every element by k.
Matrix multiplication (AB):
- Defined when cols(A) = rows(B)
- (AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ
- Not commutative: AB ≠ BA in general
- Associative: (AB)C = A(BC)
Transpose (Aᵀ): Rows become columns. Properties: (Aᵀ)ᵀ = A; (AB)ᵀ = BᵀAᵀ; (A+B)ᵀ = Aᵀ + Bᵀ
Determinants
| For 2×2: | A | = ad − bc, where A = [[a,b],[c,d]] |
|---|
For 3×3 (expansion along row 1):
| A | = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁) |
|---|
Properties of Determinants
- If two rows (or columns) are identical → determinant = 0
- Swapping two rows changes the sign
- Multiplying a row by k multiplies the determinant by k
- Adding a multiple of one row to another doesn't change the determinant
- det(AB) = det(A) × det(B)
- det(Aᵀ) = det(A)
Inverse of a Matrix
For square matrix A with det(A) ≠ 0:
A⁻¹ = adj(A) / det(A)
- Minor (Mᵢⱼ): Determinant of the submatrix obtained by deleting row i and column j
- Cofactor (Cᵢⱼ): Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ
- Adjoint (adj A): Transpose of the cofactor matrix
Solving Systems Using Cramer's Rule
For AX = B (A is 3×3, det(A) = Δ ≠ 0):
x = Δₓ/Δ, y = Δᵧ/Δ, z = Δᵤ/Δ
where Δₓ is obtained by replacing the x-column of A with B, etc.
Worked Examples
Example 1: Find the determinant of A = [[2, 3], [1, 4]].
| A | = 2×4 − 3×1 = 8 − 3 = 5 |
|---|
Example 2: Solve using Cramer's rule: x + y = 3, 2x − y = 0.
| Δ = | 1 1; 2 −1 | = −1 − 2 = −3. |
|---|---|---|
| Δₓ = | 3 1; 0 −1 | = −3. x = −3/−3 = 1. |
| Δᵧ = | 1 3; 2 0 | = −6. y = −6/−3 = 2. |
Example 3: If A = [[1, 2], [3, 4]], find A⁻¹. det(A) = 4 − 6 = −2. adj(A) = [[4, −2], [−3, 1]]. A⁻¹ = (1/−2) × [[4, −2], [−3, 1]] = [[-2, 1], [3/2, −1/2]]
Important Questions
One Mark
- Define a symmetric matrix.
- If A is 2×3 and B is 3×4, what is the order of AB?
- If |A| = 5, find |2A| for a 2×2 matrix A.
- State the condition for a matrix to be invertible.
Short Answer (3 marks)
- If A = [[1, 2], [3, 4]], verify that A(adj A) = |A| I.
- Show that the matrix [[2, 3], [1, −2]] satisfies its own characteristic equation.
- Solve the system: 2x + y = 5, 3x − 2y = 4 using Cramer's rule.
Long Answer (5 marks)
- Find the inverse of A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]] using the adjoint method.
- Solve the system x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 using Cramer's rule.
Common Mistakes
- Matrix multiplication: multiplying element-wise instead of using the row-by-column dot product.
- Cofactor signs: forgetting the (−1)^(i+j) checkerboard pattern — C₁₁ is +, C₁₂ is −, C₁₃ is +, etc.
- Adjoint: forgetting to transpose the cofactor matrix — adjoint = transpose of cofactors.
- Cramer's rule: replacing the wrong column when forming Δₓ, Δᵧ, Δᵤ.
Board Exam Tips
- "Find the inverse using adjoint" is a standard 5-mark question for 3×3 matrices — practise the cofactor calculation carefully.
- Cramer's rule for 3 unknowns is another 5-mark staple — show all three determinants clearly.
- Properties of determinants (prove |A| = 0 when rows are proportional, etc.) appear as 3-mark questions.
- JEE: Matrix equations, trace of a matrix, and inverse properties are tested regularly.