What Is a Matrix?
A matrix is a rectangular arrangement of values in rows and columns.
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▦ Matrices for Programming
Learn how rectangular arrays represent data, perform matrix operations step by step, and connect matrices with images, graphs, transformations and machine learning.
🎯 1. Learning Objectives
After completing this topic, you should be able to:
- Identify the order, rows, columns and elements of a matrix.
- Recognize common matrix types and their properties.
- Perform addition, subtraction, scalar multiplication and matrix multiplication.
- Find a transpose and the determinant of a 2 × 2 matrix.
- Check whether two matrices are compatible for an operation.
- Connect matrices with arrays, images, graphs and AI models.
🧩 2. Matrix Fundamentals
Order and Element Position
The example has 2 rows and 3 columns, so its order is 2 × 3.
Aij means the element in row i, column j. Here A23 = 7.
Programming connection: A matrix maps naturally to a two-dimensional array such as
matrix[row][column].🗂️ 3. Common Types of Matrices
Row and Column
A row matrix has one row; a column matrix has one column.
Rectangular and Square
A square matrix has equal rows and columns; otherwise it is rectangular.
Zero Matrix
Every element is 0. It acts like zero in matrix addition.
0000
Identity Matrix
Diagonal entries are 1 and all others are 0. AI = IA = A.
1001
Diagonal Matrix
All non-diagonal entries are 0.
4007
Symmetric Matrix
A square matrix is symmetric when AT = A.
2558
✅ 4. Equality, Addition and Subtraction
Equal matrices must have the same order and equal corresponding elements.
Addition and subtraction require matrices of the same order.
Operate on corresponding positions: Cij = Aij ± Bij.
Matrix addition is commutative: A + B = B + A.
1234
+
5678
=
681012
🔢 5. Scalar Multiplication
Multiply every element by the scalar value.
3 ×
1−240
=
3−6120
✖️ 6. Matrix Multiplication
Multiply each row of the first matrix by each column of the second matrix.
Compatibility rule: If A is m × n and B is n × p, then AB exists and has order m × p. The inner dimensions must match.
1234
×
5678
=
19224350
C11 = 1×5 + 2×7 = 19
C12 = 1×6 + 2×8 = 22
C21 = 3×5 + 4×7 = 43
C22 = 3×6 + 4×8 = 50
Important: Matrix multiplication is generally not commutative. Usually AB ≠ BA.
🔄 7. Transpose
The transpose exchanges rows and columns. If A is m × n, then AT is n × m.
123456
→ AT =
142536
Useful rules: (AT)T = A and (AB)T = BTAT.
📏 8. Determinant of a 2 × 2 Matrix
Formula
abcd
det(A) = ad − bc
Example
4325
det(A) = 4×5 − 3×2 = 14.
Singular matrix: If det(A) = 0, the square matrix has no inverse.
🔐 9. Inverse of a 2 × 2 Matrix
An inverse exists only when ad − bc ≠ 0.
A−1 = 1/(ad − bc)
d−b−ca
The inverse reverses a matrix transformation: AA−1 = A−1A = I.
🧪 10. Matrix Operation Explorer
Enter two 2 × 2 matrices, choose an operation and click Calculate Matrix. Results remain hidden until the button is clicked.
Matrix A
Matrix B
Result
💻 11. Matrices in Programming
Images
A grayscale image is a matrix of pixel intensities. Colour images use matrices for red, green and blue channels.
Graphs
An adjacency matrix records whether vertices are connected. Undirected graphs produce symmetric adjacency matrices.
Transformations
Rotation, scaling and reflection in computer graphics are expressed using matrix multiplication.
AI and Machine Learning
Datasets, weights and feature transformations are stored and processed as matrices.
🧾 12. Solved Problems
Problem 1: Order and Element
Problem: A has 3 rows and 4 columns. State its order and number of elements.
Solution: Order = 3 × 4; elements = 3×4 = 12.
Problem 2: Addition
Problem: Add A=[[2,1],[4,3]] and B=[[5,2],[1,6]].
Solution: Add corresponding entries: [[7,3],[5,9]].
Problem 3: Multiplication
Problem: Multiply [[1,2],[0,3]] by [[4,1],[2,5]].
Solution: Row-by-column products give [[8,11],[6,15]].
Problem 4: Determinant
Problem: Find det([[6,2],[3,4]]).
Solution: 6×4 − 2×3 = 24 − 6 = 18.
✍️ 13. Practice Problems
Solve each problem first, then use Show Solution to verify your method.
1. Find the order and number of elements in a matrix with 5 rows and 2 columns.
The order is 5 × 2, and it contains 5×2 = 10 elements.
2. Add [[3,−1],[2,5]] and [[4,6],[−2,1]].
Add corresponding elements to obtain [[7,5],[0,6]].
3. Can a 2 × 3 matrix multiply a 2 × 2 matrix in that order?
No. The inner dimensions are 3 and 2, so they do not match.
4. Find the transpose of [[1,4,7],[2,5,8]].
Rows become columns, so AT = [[1,2],[4,5],[7,8]].
5. Find the determinant of [[7,3],[2,5]] and state whether it has an inverse.
det(A) = 7×5 − 3×2 = 29. Because 29 ≠ 0, the matrix has an inverse.
📝 14. Quick Revision
- Order = number of rows × number of columns.
- Add or subtract only matrices having the same order.
- For AB, columns of A must equal rows of B.
- If A is m × n and B is n × p, then AB is m × p.
- Transpose exchanges rows and columns.
- For [[a,b],[c,d]], determinant = ad − bc.
- A 2 × 2 matrix has an inverse only when its determinant is non-zero.
- Images, graph connections and ML data can all be represented by matrices.
INTERACTIVE PRACTICE
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