Types of matrices and equality of matrices
Types include square, diagonal, scalar, identity I and zero O. Matrices are equal only with the same order and equal corresponding entries.
Types of matrices
| Type | What it means | Example |
|---|---|---|
| Row matrix | Only one row (order 1 × n) | [2, −1, 7] |
| Column matrix | Only one column (order m × 1) | [4; 0; −3] |
| Square matrix | Number of rows = number of columns | [1, 2; 3, 4] |
| Diagonal matrix | Square, with every entry off the main diagonal equal to 0 | [3, 0; 0, −2] |
| Scalar matrix | Diagonal, with all diagonal entries equal | [5, 0; 0, 5] |
| Identity matrix I | Scalar, with every diagonal entry equal to 1 | [1, 0, 0; 0, 1, 0; 0, 0, 1] |
| Zero (null) matrix O | Every entry is 0 | [0, 0, 0; 0, 0, 0] |
How the types fit together
- The main diagonal of a square matrix is a₁₁, a₂₂, a₃₃, ..., running from top left to bottom right.
- Every identity matrix is scalar, every scalar matrix is diagonal, and every diagonal matrix is square. The reverse is not true.
- [3, 0; 0, −2] is diagonal but not scalar, because its diagonal entries differ.
- A square zero matrix is also diagonal and scalar, since all its diagonal entries are equal (to 0).
- Matrices are written row by row here, with rows separated by semicolons.
Equality of matrices
Two matrices A = [aᵢⱼ] and B = [bᵢⱼ] are equal if (i) they have the same order, and (ii) each element of A equals the corresponding element of B: aᵢⱼ = bᵢⱼ for all i and j. So one matrix equation gives you one ordinary equation for each position. A 2 × 2 equation gives 4 equations, which is how board questions let you find several unknowns at once.
Example 1: find x and y
Find x and y if [x + y, 3; 5, x − y] = [8, 3; 5, 2].
- Rule: equal matrices have equal corresponding entries.
- Position (1, 1): x + y = 8 ... (1)
- Position (2, 2): x − y = 2 ... (2)
- Add (1) and (2): 2x = 10, so x = 5.
- From (1): y = 8 − 5 = 3.
x = 5, y = 3
Example 2: four unknowns (board style)
Find a, b, c and d if [2a + b, a − 2b; 5c − d, 4c + 3d] = [5, 0; 2, 13].
- Rule: equate corresponding entries.
- 2a + b = 5 ... (1) and a − 2b = 0 ... (2)
- From (1), b = 5 − 2a. Put this in (2): a − 10 + 4a = 0, so 5a = 10 and a = 2. Then b = 1.
- 5c − d = 2 ... (3) and 4c + 3d = 13 ... (4)
- From (3), d = 5c − 2. Put this in (4): 4c + 15c − 6 = 13, so 19c = 19 and c = 1. Then d = 3.
- Check: 4 + 1 = 5, 2 − 2 = 0, 5 − 3 = 2, 4 + 9 = 13.
a = 2, b = 1, c = 1, d = 3
Same numbers, not equal
They hold the same numbers, but their orders differ, so they are not equal. Always check the order first.
Photo editors start a blank black canvas as a zero matrix (every pixel value 0). Multiplying by an identity matrix is the 'do nothing' step: it leaves a picture exactly as it was.
1 marks: “A 3 × 3 diagonal matrix has every diagonal entry equal to 4. What type of matrix is it, and how can it be written using I?” It is a scalar matrix, equal to 4I.
2 marks: “Find x and y if [2x − y, 5; 3, y] = [6, 5; 3, −2].” From position (2, 2), y = −2. From position (1, 1), 2x − y = 6, so 2x + 2 = 6 and x = 2.
Same shape, same spots, same numbers: that is equality. Identity ⊂ scalar ⊂ diagonal ⊂ square.