Matrices
Rectangular arrays of numbers that encode whole systems at once — the language of linear algebra
- The order and notation of a matrix, and the main types you must recognise on sight.
- Addition, scalar multiplication, and the rules they obey.
- Matrix multiplication — the row-by-column rule and why it is not commutative.
- The transpose and its laws, especially \((AB)^{T}=B^{T}A^{T}\).
- Symmetric and skew-symmetric matrices, and the decomposition of any square matrix.
- The inverse via the adjoint, and its key properties.
What Is a Matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with \(m\) rows and \(n\) columns has order \(m\times n\), and its entry in row \(i\), column \(j\) is written \(a_{ij}\). The whole matrix is then \(A=[a_{ij}]_{m\times n}\). Matrices matter because a single array can carry an entire system of equations, a transformation of space, or a table of data — and we can compute with the array as one object.
Types of Matrices
A handful of special shapes recur so often they have names. Recognising them instantly saves time on nearly every problem.
| Name | Defining condition |
|---|---|
| Row / Column | order \(1\times n\) / order \(m\times1\) |
| Square | \(m=n\) (order \(n\)) |
| Diagonal | square, \(a_{ij}=0\) for \(i\neq j\) |
| Scalar | diagonal with all diagonal entries equal |
| Identity \(I\) | scalar with each diagonal entry \(1\) |
| Null / Zero \(O\) | every entry \(0\) |
| Triangular | upper: \(a_{ij}=0\) for \(i>j\); lower: for \(i |
Equality, Addition & Scalar Multiplication
The simplest operations act entry by entry, and they require the matrices to match in shape.
Two matrices are equal only if they have the same order and every corresponding entry agrees. Addition needs the same order; scalar multiplication multiplies every entry by \(k\). Addition is commutative and associative, exactly like numbers.
Matrix Multiplication
Multiplication is where matrices stop behaving like numbers. To form \(AB\), the number of columns of \(A\) must equal the number of rows of \(B\). The \((i,j)\) entry of the product is the "dot product" of row \(i\) of \(A\) with column \(j\) of \(B\).
If \(A\) is \(m\times p\) and \(B\) is \(p\times n\), then \(AB\) is \(m\times n\). It is associative and distributive, but not commutative: in general \(AB\neq BA\), and \(BA\) may not even be defined.
Matrix multiplication breaks two habits from ordinary algebra. First, \(AB=O\) does not force \(A=O\) or \(B=O\) — there are zero divisors. Second, \(AB=AC\) does not allow you to cancel \(A\) unless \(A\) is invertible. Treat order and cancellation with care.
The Transpose
The transpose \(A^{T}\) flips a matrix across its main diagonal — rows become columns. If \(A\) is \(m\times n\), then \(A^{T}\) is \(n\times m\) with \((A^{T})_{ij}=a_{ji}\).
| Law | Statement |
|---|---|
| Involution | \((A^{T})^{T}=A\) |
| Sum | \((A+B)^{T}=A^{T}+B^{T}\) |
| Scalar | \((kA)^{T}=k\,A^{T}\) |
| Product (reversal) | \((AB)^{T}=B^{T}A^{T}\) |
Symmetric & Skew-Symmetric Matrices
Two classes of square matrix are defined by how they relate to their own transpose, and together they split every square matrix cleanly in two.
A skew-symmetric matrix has zero diagonal (since \(a_{ii}=-a_{ii}\)). Every square matrix splits uniquely as \(A=\underbrace{\tfrac12(A+A^{T})}_{\text{symmetric}}+\underbrace{\tfrac12(A-A^{T})}_{\text{skew}}\).
The Inverse of a Matrix
Division has no direct analogue for matrices; instead we look for an inverse \(A^{-1}\) with \(AA^{-1}=A^{-1}A=I\). Only square matrices can have one, and only when they are non-singular.
The matrix is invertible iff \(\det A\neq0\) (non-singular). When it exists the inverse is unique, and the product rule reverses: \((AB)^{-1}=B^{-1}A^{-1}\). Also \((A^{T})^{-1}=(A^{-1})^{T}\).
For \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\) with \(ad-bc\neq0\), the inverse is \(A^{-1}=\dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\) — swap the diagonal, negate the off-diagonal, divide by the determinant. Memorise this one; it appears constantly.
Elementary Row Operations
Three reversible moves on the rows of a matrix let you simplify it without changing what it represents — they are the engine behind the Gauss–Jordan method for inverses and systems of equations.
Interchange two rows: \(R_i\leftrightarrow R_j\).
Multiply a row by a non-zero scalar: \(R_i\to kR_i\).
Add a multiple of one row to another: \(R_i\to R_i+kR_j\).
Trace & Other Special Matrices
A few more definitions round out the vocabulary and appear regularly in problems.
| Term | Meaning |
|---|---|
| Trace \(\operatorname{tr}A\) | sum of the diagonal entries; \(\operatorname{tr}(AB)=\operatorname{tr}(BA)\) |
| Orthogonal | \(AA^{T}=I\), i.e. \(A^{T}=A^{-1}\) |
| Idempotent | \(A^{2}=A\) |
| Nilpotent | \(A^{k}=O\) for some \(k\) |
| Involutory | \(A^{2}=I\) |
Strategies & Standard Results
A short field guide for matrix problems.
Before multiplying, confirm columns of the first equal rows of the second; note the order of the result.
Never assume \(AB=BA\); keep left and right factors fixed when manipulating.
To decide invertibility, compute \(\det A\); non-zero means an inverse exists.
Putting It to Work
Problem. With \(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\) and \(B=\begin{bmatrix}0&-1\\5&2\end{bmatrix}\), find \(2A-B\).
Problem. For the \(A,B\) above, show \(AB\neq BA\).
The two products differ entirely — order matters.
Problem. Express \(A=\begin{bmatrix}2&3\\1&4\end{bmatrix}\) as a symmetric plus a skew-symmetric matrix.
Problem. Find the inverse of \(A=\begin{bmatrix}2&5\\1&3\end{bmatrix}\).
Solution. \(\det A=2\cdot3-5\cdot1=1\), so the shortcut gives
Problem. If \(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\) and \(AX=\begin{bmatrix}5\\11\end{bmatrix}\), find \(X\).
Solution. \(\det A=-2\), so \(A^{-1}=-\tfrac12\begin{bmatrix}4&-2\\-3&1\end{bmatrix}\). Then \(X=A^{-1}\begin{bmatrix}5\\11\end{bmatrix}\):
Problem. If \(A=\begin{bmatrix}1&1\\0&1\end{bmatrix}\), find \(A^{n}\).
Solution. Computing a few powers reveals the pattern, confirmed by induction:
Chapter Summary
Order \(m\times n\), entry \(a_{ij}\); know square, diagonal, scalar, identity, null, triangular.
Add/scale entrywise (same order); multiply row-by-column when inner orders match.
Associative, distributive, not commutative; \(AB=O\) need not give \(A=O\) or \(B=O\).
\((AB)^{T}=B^{T}A^{T}\); symmetric \(A^{T}=A\), skew \(A^{T}=-A\).
\(A^{-1}=\tfrac{1}{\det A}\operatorname{adj}A\), exists iff \(\det A\neq0\); \((AB)^{-1}=B^{-1}A^{-1}\).
Orthogonal \(AA^T=I\), idempotent \(A^2=A\), involutory \(A^2=I\).
Problems
For each, check orders before operating. Difficulty rises down the list.
- If \(A=\begin{bmatrix}1&-1\\2&3\end{bmatrix}\) and \(B=\begin{bmatrix}2&0\\1&-2\end{bmatrix}\), compute \(3A-2B\) and \(AB\).
- Find \(x,y\) so that \(\begin{bmatrix}x+y&2\\3&x-y\end{bmatrix}=\begin{bmatrix}5&2\\3&1\end{bmatrix}\).
- Show that \(A=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\) is orthogonal, and find \(A^{-1}\).
- Express \(\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}\) as a symmetric plus a skew-symmetric matrix.
- If \(A\) is symmetric and \(B\) is skew-symmetric of the same order, what can you say about \(A+B\) and \(AB-BA\)?
- Verify \((AB)^{T}=B^{T}A^{T}\) for two \(2\times2\) matrices of your choice.
- Find the inverse of \(\begin{bmatrix}1&2&3\\0&1&4\\0&0&1\end{bmatrix}\) by elementary row operations.
- If \(A^{2}=A\) (idempotent), simplify \((I+A)^{3}-7A\).
- Solve the system \(x+2y=3,\ 3x+4y=11\) by writing it as \(AX=B\) and inverting \(A\).
- For \(A=\begin{bmatrix}2&-1\\1&0\end{bmatrix}\), show that \(A^{2}=2A-I\) and hence find \(A^{4}\).
- If \(\operatorname{tr}(A)=0\) for a \(2\times2\) matrix with \(A^2=I\), what are the possible values of \(\det A\)?
- Prove that the product of two lower-triangular matrices is lower-triangular.