- Chapter 3
- Algebra
- 24 questions
Matrices
Order, types, operations, transpose, symmetric and skew-symmetric matrices, and invertibility.
A matrix is an array of numbers treated as a single object. That sounds like bookkeeping, and at first it is — but once multiplication is defined, a matrix stops being a table and becomes an operation.
The surprise of this chapter is how much ordinary arithmetic fails to carry over. Matrix multiplication is not commutative: AB and BA are usually different, and may not both even exist. A product can be the zero matrix without either factor being zero. Working carefully with these differences, rather than assuming familiar rules still hold, is most of what the chapter asks.
What you should be able to do
- Classify matrices by order and type, and test two matrices for equality.
- Add, subtract, scalar-multiply and multiply matrices, checking order compatibility first.
- Use the properties of the transpose, and express a square matrix as the sum of a symmetric and a skew-symmetric matrix.
- Explain what makes a matrix invertible, and why the inverse is unique when it exists.
Topics
4 topics
- Concept, Notation, Order and Types of Matrices
Order and notation, equality of matrices, and the standard types including zero and identity matrices.
- Order and notation
- Equality of matrices
- Row, column, square, diagonal, scalar, zero and identity matrices
- Operations on Matrices
Addition, scalar multiplication and matrix multiplication, with their properties and the failure of commutativity.
- Addition and scalar multiplication
- Multiplication of matrices
- Non-commutativity and zero divisors
- Transpose, Symmetric and Skew-Symmetric Matrices
Properties of the transpose, and decomposing a square matrix into symmetric and skew-symmetric parts.
- Invertible Matrices
The definition of an inverse and the uniqueness of the inverse when it exists.
Formulas
10 items · All formulas
Order of a matrix product
A_{m \times n} \, B_{n \times p} = (AB)_{m \times p}The product exists only when the columns of A match the rows of B; the outer dimensions give the result's order.
Entry of a product
(AB)_{ij} = \sum_{k=1}^{n} a_{ik}\,b_{kj}Row i of A paired term by term with column j of B.
Transpose of a product
(AB)^{T} = B^{T} A^{T}The order reverses — this is forced by the dimension requirement.
Transpose properties
(A^{T})^{T} = A, \quad (A + B)^{T} = A^{T} + B^{T}, \quad (kA)^{T} = k\,A^{T}Symmetric matrix
A^{T} = A \quad \text{i.e.} \quad a_{ij} = a_{ji}Entries mirror across the main diagonal.
- Valid when
- Defined only for square matrices.
Skew-symmetric matrix
A^{T} = -A \quad \text{i.e.} \quad a_{ij} = -a_{ji}Forces a_{ii} = 0: every diagonal entry of a skew-symmetric matrix is zero.
Symmetric–skew decomposition
A = \tfrac{1}{2}\left(A + A^{T}\right) + \tfrac{1}{2}\left(A - A^{T}\right)The unique way to write a square matrix as symmetric plus skew-symmetric.
- Valid when
- A square.
Inverse of a product
(AB)^{-1} = B^{-1} A^{-1}Like the transpose, the order reverses.
- Valid when
- A and B both invertible and of the same order.
Definition of the inverse
AB = BA = I \;\Rightarrow\; B = A^{-1}The inverse must work on both sides; when it exists it is unique.
- Valid when
- A square.
Number of entries
\text{An } m \times n \text{ matrix has } mn \text{ entries.}Used in counting questions: the possible orders of a matrix with mn entries correspond to the factor pairs of mn.
Key points
8 items · All key points
Matrix multiplication is not commutative: AB \neq BA in general, and often only one product is even defined.
Always check orders first. If A is 2\times3 and B is 3\times2, then AB is 2\times2 but BA is 3\times3 — different matrices entirely.
AB = O does not imply A = O or B = O.
Matrices have zero divisors, unlike real numbers. This is why factorising a matrix equation to zero proves nothing on its own.
Cancellation is invalid: AB = AC does not give B = C unless A is invertible.
Every diagonal entry of a skew-symmetric matrix is zero.
From a_{ii} = -a_{ii} we get 2a_{ii} = 0. Use it as an instant check on your skew-symmetric part.
(AB)^{T} = B^{T}A^{T} and (AB)^{-1} = B^{-1}A^{-1} — both reverse the order.
Addition requires the two matrices to have exactly the same order; multiplication does not.
This asymmetry is the source of many order errors — the rules for the two operations are genuinely different.
The identity matrix I satisfies AI = IA = A — it is the one matrix that does commute with everything of its order.
Two matrices are equal only when they have the same order and every corresponding entry matches.
Common mistakes
6 items · All common mistakes
- Mistake
Assuming AB = BA and simplifying an expression on that basis.
InsteadExpand (A+B)^2 as A^2 + AB + BA + B^2. It equals A^2 + 2AB + B^2 only if A and B commute.
WhyThe identity (a+b)^2 = a^2 + 2ab + b^2 is so automatic from ordinary algebra that the middle terms get combined without thought.
- Mistake
Concluding from AB = O that A = O or B = O.
InsteadNot valid for matrices. Non-zero matrices can multiply to the zero matrix.
WhyReal numbers have no zero divisors, so the habit transfers wrongly.
- Mistake
Writing (AB)^{T} = A^{T}B^{T}.
Instead(AB)^{T} = B^{T}A^{T} — the order reverses.
WhyThe transpose distributes over addition without reordering, so the same is assumed for products.
- Mistake
Attempting to multiply matrices whose inner dimensions do not match.
InsteadFor AB, the number of columns of A must equal the number of rows of B. Write the orders down before starting.
WhyUnder time pressure students begin computing entries without the compatibility check.
- Mistake
Producing a 'skew-symmetric part' with non-zero entries on the diagonal.
InsteadRecompute — the diagonal of \frac{1}{2}(A - A^T) is always zero, so a non-zero entry there signals an arithmetic error.
- Mistake
Forgetting the factor \frac{1}{2} when splitting A into symmetric and skew parts.
InsteadWithout it the two parts add to 2A. Verify by adding them back together.