Matrices and Determinants
Matrices: Order, Equality, Addition and Scalar Multiplication
What a matrix is, how its entries are indexed, when two are equal, and the arithmetic that works entry by entry.
What this page covers
- State the order of a matrix and index its entries
- Decide whether two matrices are equal
- Add and subtract matrices of the same order
- Multiply a matrix by a scalar
What a matrix is
A rectangular array of numbers. Its order is the number of rows by the number of columns.
The source introduces the object with a memorable mispronunciation note, then gets straight to the notation: a matrix A = (aij) has order rows × columns.
Indexing — the source's example
| 1 | -3 |
| 2 | 1 |
a21 is row 2, column 1 — the entry 2 here. Reversing the convention is the commonest indexing error, and for a non-square matrix it produces an entry that does not exist.
| Name | Condition | Example order |
|---|---|---|
| Square | Rows equal columns | 2 × 2 |
| Row matrix (row vector) | One row | 1 × 3 |
| Column matrix (column vector) | One column | 3 × 1 |
| Zero matrix | Every entry zero | Any |
The source notes that a column matrix is also called a column vector, which is how a system's unknowns and constants are written when it is put in the form Ax = b.
Equality
Two matrices are equal if and only if they have the same order and aij = bij for every i and j.
The source's example makes the order condition concrete:
| 1 | -2 | 4 |
| 2 | 1 | 6 |
is not equal to
| 1 | 2 |
| -3 | 1 |
| 4 | 6 |
even though the entries look related. Different orders means not equal, whatever the entries.
Addition and subtraction
C = A + B if and only if cij = aij + bij for all i, j. Both matrices must have the same order.
Worked example — the source's case
For
| 1 | -3 |
| 2 | 1 |
and
| 4 | 7 |
| 2 | -3 |
the sum is
| 5 | 4 |
| 4 | -2 |
Subtraction works the same way: A - B has entries aij - bij. Equivalently it is A + (-B), where -B is B with every sign reversed.
A combined operation — the source's case
With B and C as given, compute 2B - C.
First 2B:
| 4 | -6 |
| 8 | 4 |
Then subtract C = [[7, 2], [-1, 4]]:
| -3 | -8 |
| 9 | 0 |
Scalar multiplication
If B = kA for a scalar k, then bij = k aij for every entry. Every entry is multiplied.
The source's example
Doubling A = [[1, -3], [2, 1]]:
| 2 | -6 |
| 4 | 2 |
The matrix -A is the special case k = -1. The source computes it and verifies the inverse property:
| -1 | 3 |
| -2 | -1 |
Scalar multiplication touches every entry. This differs from multiplying a determinant by a scalar, where only one row need be scaled — a distinction worth keeping in mind later.
The algebra of matrix addition
The source lists the laws for matrices of the same order, and they are exactly the field laws for addition.
| Law | Statement | Note |
|---|---|---|
| Commutative | A + B = B + A | Holds — unlike matrix multiplication |
| Associative | A + (B + C) = (A + B) + C | Holds |
| Identity | A + O = A | O is the zero matrix of the same order |
| Additive inverse | A + (-A) = O | Always exists |
| Scalar distributive | k(A + B) = kA + kB | Holds |
Every one holds. It is matrix multiplication where the laws start to fail, and that failure is the subject of Matrix Multiplication and Matrix Algebra.
The same five laws hold for vector addition, as recorded in Week 10. Matrices, vectors and real numbers all share this structure under addition, which is why so much notation transfers between them.
Common mistakes
| Mistake | Correct | Why |
|---|---|---|
| a21 read as row 1, column 2 | Row index first | a21 is row 2, column 1 |
| Adding matrices of different orders | Orders must match | Every entry needs a partner |
| Order given as columns by rows | Rows first | 2 × 3 means two rows |
| Scaling only the first row | Every entry is multiplied | Unlike a determinant |
| Calling two matrices equal because the entries match in a different arrangement | Order must match too | 2 × 3 ≠ 3 × 2 |
| Assuming multiplication behaves like addition | It does not commute | See the multiplication page |
Frequently asked questions
What is the order of a matrix?
Rows by columns, in that order. A 2 × 3 matrix has two rows and three columns. The source is explicit: number of rows × number of columns.
When are two matrices equal?
When they have the same order and every corresponding entry matches. The source states both conditions; the first alone is not enough.
Can I add matrices of different orders?
No. Addition is entry by entry, so every entry must have a partner. A 2 × 3 and a 3 × 2 matrix cannot be added.
What does aij mean?
The entry in row i and column j. Row index first, always. So a21 is in the second row, first column.
Source. Handwritten teaching notes, Week 11, pages 8-11.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.