Unit 1: Matrix algebra
Mathematics-II notes · PTU syllabus (BSIT202/BSBC202)
On this page
- Unit summary
- Types of matrices
- Matrix addition, subtraction and scalar multiplication
- Multiplication of matrices
- Transpose of a matrix
- Determinants, minors and cofactors
- Adjoint and inverse of a matrix
- Elementary transformations and rank
- Solving simultaneous equations
- Key terms
- Quick revision
- Important questions
Unit summary
Matrices organise data and solve systems of equations — the basis of graphics, networks and machine learning. This unit covers types of matrices and operations, determinants, minors, cofactors, adjoint and inverse of a matrix, elementary transformations, rank of a matrix, and solving simultaneous equations by Cramer's rule and matrix inversion.
After this unit you can
- Identify types of matrices and perform matrix operations
- Evaluate determinants, minors and cofactors
- Find the adjoint, inverse and rank of a matrix
- Solve simultaneous equations using Cramer's rule and matrix inversion
PTU syllabus topics
- Types of matrices and operations
- determinants (without properties)
- minors
- cofactors
- adjoint and inverse of a matrix
- elementary transformations
- rank of a matrix
- solving simultaneous equations using Cramer's rule and matrix inversion
Inverse
A⁻¹ = adj(A) / det(A)
Exists only if det(A) ≠ 0
Cramer's rule
x = Dx / D, y = Dy / D
Matrix method
X = A⁻¹ B for AX = B
Rank
Order of the largest non-zero minor
Topic 1
Types of matrices
| Type | Definition | Example |
|---|---|---|
| Row matrix | Only one row (1 × n) | [3 1 4] |
| Column matrix | Only one column (m × 1) | [2; 5; 9] |
| Rectangular matrix | Rows ≠ columns | 2 × 3 matrix |
| Square matrix | Rows = columns | 3 × 3 matrix |
| Diagonal matrix | Square; all non-diagonal elements are 0 | [4 0; 0 7] |
| Scalar matrix | Diagonal matrix with all diagonal elements equal | [5 0; 0 5] |
| Unit (identity) matrix | Scalar matrix with diagonal elements 1, written I | [1 0; 0 1] |
| Null (zero) matrix | Every element is 0, written O | [0 0; 0 0] |
- Comparable matrices have the same order.
- Equal matrices have the same order and equal corresponding elements.
Exam tip
Every unit matrix is a scalar matrix, and every scalar matrix is a diagonal matrix — but not the other way round.
Topic 2
Matrix addition, subtraction and scalar multiplication
- Scalar multiplication: multiply every element by the number k. If A = [1 2; 3 4], then 3A = [3 6; 9 12].
- Negative of a matrix: −A = (−1)A, so every element changes sign.
- Addition and subtraction: only for matrices of the same order; add or subtract corresponding elements.
Example
A = [2 3; 1 4], B = [5 1; 0 2]. A + B = [7 4; 1 6] and A − B = [−3 2; 1 2].
Properties of addition: A + B = B + A (commutative), (A + B) + C = A + (B + C) (associative), A + O = A, and A + (−A) = O.
Topic 3
Multiplication of matrices
The product AB is defined only when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p.
- 1Check orders
Columns of A must equal rows of B
- 2Pick row i of A
- 3Pick column j of B
- 4Multiply pairwise and add
This gives element (i, j) of AB
- 5Repeat for every row and column
Example
A = [1 2; 3 4], B = [5 6; 7 8]. AB = [1·5 + 2·7 1·6 + 2·8; 3·5 + 4·7 3·6 + 4·8] = [19 22; 43 50].
- Matrix multiplication is not commutative: in general AB ≠ BA.
- It is associative: (AB)C = A(BC), and distributive: A(B + C) = AB + AC.
- AI = IA = A for the identity matrix I.
Exam tip
Always write "Since A is m × n and B is n × p, AB exists and is of order m × p" before multiplying. Examiners give marks for it.
Topic 4
Transpose of a matrix
The transpose of A, written Aᵀ or A', is obtained by changing rows into columns. If A is m × n, Aᵀ is n × m.
Example
A = [1 2 3; 4 5 6] (2 × 3). Aᵀ = [1 4; 2 5; 3 6] (3 × 2).
Properties:
- (Aᵀ)ᵀ = A
- (A + B)ᵀ = Aᵀ + Bᵀ
- (kA)ᵀ = kAᵀ
- (AB)ᵀ = BᵀAᵀ (note the reversed order)
- A square matrix is symmetric if Aᵀ = A and skew-symmetric if Aᵀ = −A.
Topic 5
Determinants, minors and cofactors
2 × 2
det [a b; c d] = ad − bc
3 × 3 (expansion along row 1)
a11 C11 + a12 C12 + a13 C13
Minor Mij
Determinant left after deleting row i and column j
Cofactor Cij
(−1)^(i+j) × Mij
Example
A = [2 1 3; 0 4 5; 1 0 6]: det A = 2(24 − 0) − 1(0 − 5) + 3(0 − 4) = 48 + 5 − 12 = 41.
Topic 6
Adjoint and inverse of a matrix
Adjoint
Transpose of the matrix of cofactors
Inverse
A⁻¹ = adj(A) ÷ det(A), when det(A) ≠ 0
2 × 2 shortcut
[a b; c d]⁻¹ = (1 ÷ (ad − bc)) [d −b; −c a]
Check
A × A⁻¹ = I
- A matrix with determinant 0 is singular and has no inverse.
Example
A = [4 7; 2 6]: det = 24 − 14 = 10; A⁻¹ = (1/10) [6 −7; −2 4].
Topic 7
Elementary transformations and rank
- Elementary row operations: interchange two rows; multiply a row by a non-zero number; add a multiple of one row to another (similarly for columns).
- Rank: the order of the largest non-zero minor, or the number of non-zero rows in row-echelon form.
- 1Apply row operations to get zeros below each leading entry
- 2Reach row-echelon form
- 3Count non-zero rows = rank
- Use: consistency of linear equations — the system is consistent if rank of the coefficient matrix equals rank of the augmented matrix.
Topic 8
Solving simultaneous equations
x
Dx ÷ D
y
Dy ÷ D
Condition
D ≠ 0, where D is the coefficient determinant
Example
2x + 3y = 8, x − y = −1: D = −2 − 3 = −5; Dx = (8)(−1) − (3)(−1) = −5; Dy = 2(−1) − 8(1) = −10; x = 1, y = 2.
- Matrix inversion method: write AX = B; then X = A⁻¹B (requires det A ≠ 0).
Key terms
- Singular matrix
- Matrix with zero determinant
- Cofactor
- Signed minor
- Adjoint
- Transpose of the cofactor matrix
- Rank
- Maximum number of linearly independent rows
- Cramer's rule
- Solving equations using determinants
Quick revision
- Types of matrices; addition, scalar multiplication, multiplication, transpose.
- Determinants of 2 × 2 and 3 × 3; minors and cofactors.
- adj(A); A⁻¹ = adj(A)/|A|; singular matrices.
- Elementary operations; echelon form; rank; consistency.
- Cramer's rule; X = A⁻¹B.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.Define a symmetric matrix.
- Q2.Find the determinant of [3 2; 1 4].
- Q3.What is a cofactor?
- Q4.When does a matrix have no inverse?
- Q5.Define the rank of a matrix.
- Q6.State Cramer's rule.
Long-answer questions
- Q1.Find the inverse of a 3 × 3 matrix using the adjoint method (numerical).
- Q2.Find the rank of a matrix by elementary transformations (numerical).
- Q3.Solve a system of three equations by Cramer's rule (numerical).
- Q4.Solve simultaneous equations by the matrix inversion method (numerical).
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