Mathematics — Algebra

Subtopic: MATRICES (for learners around 14 years)

A matrix is a rectangular arrangement of numbers in rows and columns. We write the size of a matrix as m × n (m rows and n columns). Each entry is usually written as a_ij where i is the row number and j is the column number.

Notation and a simple example

This is a 2 × 2 matrix. We say A = [a_ij] where a₁₁=1, a₁₂=2, a₂₁=3, a₂₂=4.

Common types of matrices

• Row matrix (1 × n), column matrix (m × 1).
• Square matrix (n × n).
• Zero matrix (all entries 0).
• Diagonal matrix (non-zero on main diagonal only).
• Identity matrix I_n (1 on diagonal, 0 elsewhere).
• Symmetric (A = A^T), skew-symmetric (A = −A^T).

Basic operations

Equality

Two matrices are equal if they have the same size and each corresponding entry is equal.

Addition and subtraction

You can add or subtract two matrices only when they have the same size. Add/subtract corresponding entries.

Scalar multiplication

Multiply every entry by the scalar.

3A =

3	6
9	12

Matrix multiplication

You can multiply A (m × n) by B (n × p). The result is an m × p matrix. Multiply rows of A by columns of B and add products.

Transpose

The transpose of A, written A^T, is formed by swapping rows and columns. Example:

Determinant (for 2 × 2 matrices)

For matrix M = [[a, b], [c, d]], det(M) = ad − bc.

Example: For A = [[1,2],[3,5]], det(A) = 1·5 − 2·3 = 5 − 6 = −1.

Inverse of a 2 × 2 matrix (if det ≠ 0)

If A = [[a, b], [c, d]] and det(A) ≠ 0, then

A⁻¹ = (1 / det(A)) · [[d, −b], [−c, a]].

Example: For A = [[1,2],[3,5]], det = −1, so

A⁻¹ = (1/−1) · [[5, −2], [−3, 1]] = [[−5, 2], [3, −1]].

Using matrices to solve a system of linear equations

Write the system in the form A x = b, then x = A⁻¹ b (if inverse exists).

Example:

x + 2y = 5
3x + 5y = 11

Here A = [[1,2],[3,5]], x = [[x],[y]], b = [[5],[11]]. We found A⁻¹ = [[−5,2],[3,−1]].

Now x = A⁻¹ b = [[−5,2],[3,−1]] · [[5],[11]] =

-5·5 + 2·11 = -25 + 22 = -3

3·5 − 1·11 = 15 − 11 = 4

So x = −3, y = 4.

Exam tips and checks

• Always write the size of a matrix, e.g. 2 × 3.
• Check dimensions before adding or multiplying.
• For 2 × 2 inverse, check det ≠ 0 first.
• Practice showing rows × columns when multiplying so marks are clear on tests.
• Use matrices in word problems (e.g. simple economics or mixing problems) — KCSE-style questions often test method and clear working.

Practice questions (try on your own)

1. Compute A + B for A = [[1, 0],[2, 3]] and B = [[4, 5],[1, 2]].
2. Find A × B for A = [[1, 2],[0, 1]] and B = [[3, 1],[2, 4]].
3. Solve by matrices: x + y = 4 and 2x − y = 1.

Answers

1. A + B = [[5, 5],[3, 5]].
2. A × B =

   7	9
   2	4

3. Write A = [[1,1],[2,−1]], b = [[4],[1]]. det(A) = 1·(−1) − 1·2 = −3 ≠ 0. A⁻¹ = (1/−3) · [[−1, −1], [−2, 1]] = [[1/3, 1/3], [2/3, −1/3]]. Multiply A⁻¹b to get x = 1, y = 3.

If you want, I can make printable A4 notes, more worked examples, or add step-by-step animated calculations for any example — tell me which.