Matrices as data tables
A matrix is simply an organised table. Rows and columns carry meaning – e.g. rows = shops, columns = items – and matrix operations respect that meaning.
Chapter Overview
A matrix is a rectangular arrangement of numbers that lets us handle many quantities at once – stock in several shops, marks in several subjects, or coefficients of several equations. In this chapter you will learn the language of matrices, how to add and multiply them, the transpose, symmetric and skew-symmetric matrices, and what it means for a matrix to be invertible.
Why it matters: Matrices are the foundation for Determinants (next chapter), solving systems of equations, and much of modern computing, graphics and data science.
Topics
Key Concepts
- Order \(m\times n\); element \(a_{ij}\) in row \(i\), column \(j\).
- \(AB\) exists only when (columns of A) = (rows of B); in general \(AB \neq BA\).
- \((AB)^T = B^TA^T\) and \((AB)^{-1} = B^{-1}A^{-1}\).
- Every square matrix = symmetric part + skew-symmetric part.
Formulas
Number of entries
Matrices · Matrices and Their Types\[\text{order } m\times n \;\Rightarrow\; mn \text{ entries}\]
Possible orders of a matrix with \(N\) entries correspond to factor pairs of \(N\). E.g. 12 entries: \(1\times12, 2\times6, 3\times4, 4\times3, 6\times2, 12\times1\).
Entry of a product
Matrices · Operations on Matrices\[(AB)_{ij} = \sum_{k=1}^{n} a_{ik}\,b_{kj}\]
Multiply row \(i\) of \(A\) by column \(j\) of \(B\) term by term and add. Requires columns of \(A\) = rows of \(B\).
Order of a product
Matrices · Operations on Matrices\[(m\times n)(n\times p) = (m\times p)\]
The inner dimensions must match; the outer ones give the order of the product.
Reversal law for transpose
Matrices · Transpose of a Matrix\[(AB)^{T} = B^{T}A^{T}\]
Also \((A^T)^T = A\), \((A+B)^T = A^T + B^T\), \((kA)^T = kA^T\).
Properties of transpose
Matrices · Transpose of a Matrix\[(A^T)^T=A,\quad (kA)^T=kA^T,\quad (A+B)^T=A^T+B^T,\quad (AB)^T=B^TA^T\]
Symmetric + skew-symmetric split
Matrices · Symmetric and Skew-Symmetric Matrices\[A = \tfrac12(A + A^{T}) + \tfrac12(A - A^{T})\]
The first part is symmetric, the second skew-symmetric. Works for every square matrix.
Inverse of a 2 × 2 matrix
Matrices · Invertible Matrices\[\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\]
Valid only when \(ad - bc \ne 0\). Swap the diagonal, negate the off-diagonal, divide by the determinant.
Reversal law for inverse
Matrices · Invertible Matrices\[(AB)^{-1} = B^{-1}A^{-1}\]
Also \((A^{-1})^{-1} = A\) and \((A^T)^{-1} = (A^{-1})^T\).
Key Points
Order is rows × columns
Matrices · Matrices and Their TypesAlways state order as (rows) × (columns). A \(2\times3\) and a \(3\times2\) matrix are different.
AB ≠ BA in general
Matrices · Operations on MatricesNever assume matrices commute. Keep the order of factors in every step.
Even when both \(AB\) and \(BA\) exist, they may differ – and one may exist while the other does not.
No cancellation law
Matrices · Operations on Matrices\(AB = AC\) does not imply \(B = C\), and \(AB = O\) does not imply \(A = O\) or \(B = O\).
Skew-symmetric ⇒ zero diagonal
Matrices · Symmetric and Skew-Symmetric MatricesFrom \(a_{ii} = -a_{ii}\), every diagonal entry of a skew-symmetric matrix is 0.
Counting free entries
Matrices · Symmetric and Skew-Symmetric MatricesAn \(n\times n\) symmetric matrix is fixed by \(\frac{n(n+1)}{2}\) entries; a skew-symmetric one by \(\frac{n(n-1)}{2}\) entries (its diagonal is zero).
Only square matrices have inverses
Matrices · Invertible MatricesInverse needs AB = BA = I, which requires square matrices of the same order.
Using a matrix equation to find the inverse
Matrices · Invertible MatricesIf \(A^2-3A+2I=O\) then \(A(3I-A)=2I\), so \(A^{-1}=\tfrac12(3I-A)\). Multiply the equation by \(A^{-1}\) whenever the constant term is non-zero.
Common Mistakes
Squaring entries instead of the matrix
Matrices · Operations on MatricesWriting \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}^2 = \begin{bmatrix} 4 & 1 \\ 0 & 9 \end{bmatrix}\).
Correct: \(A^2 = A\cdot A = \begin{bmatrix} 4 & 5 \\ 0 & 9 \end{bmatrix}\). Matrix powers use row × column multiplication.
Using (A + B)² = A² + 2AB + B²
Matrices · Operations on MatricesApplying the number identity to matrices.
Correct: \((A+B)^2 = A^2 + AB + BA + B^2\). It reduces to the familiar form only if \(AB = BA\).
Treating matrix algebra like numbers
Matrices · Operations on Matrices\(AB=O\) does not force \(A=O\) or \(B=O\), and \((A+B)^2=A^2+AB+BA+B^2\), not \(A^2+2AB+B^2\), unless \(AB=BA\).
Not reversing the order in (AB)ᵀ
Matrices · Transpose of a MatrixWriting \((AB)^T = A^TB^T\).
Correct: \((AB)^T = B^TA^T\). Check with orders: if \(A\) is \(2\times3\) and \(B\) is \(3\times4\), \(A^TB^T\) is not even defined.
Half-applying the 2 × 2 inverse rule
Matrices · Invertible MatricesSwapping the diagonal entries but forgetting to change the signs of the off-diagonal entries (or vice versa).
Correct: swap \(a\) and \(d\), change the signs of \(b\) and \(c\), then divide by \(ad - bc\). Verify with \(AA^{-1} = I\).
Solved Examples
Checking whether AB = BA
Matrices · Operations on MatricesFor \(A = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 2 & 1 \\ -1 & 4 \end{bmatrix}\), find \(AB\) and \(BA\). Are they equal?
- \(AB = \begin{bmatrix} 4 & -7 \\ 6 & 3 \end{bmatrix}\).
- \(BA = \begin{bmatrix} 5 & -4 \\ 11 & 2 \end{bmatrix}\).
- Compare corresponding entries: they differ.
Final Answer: \(AB \ne BA\): matrix multiplication is not commutative.
Splitting a 3 × 3 matrix
Matrices · Symmetric and Skew-Symmetric MatricesExpress \(A = \begin{bmatrix} 1 & 2 & 4 \\ 6 & 8 & 1 \\ 3 & 5 & 7 \end{bmatrix}\) as the sum of a symmetric and a skew-symmetric matrix.
- \(P = \tfrac12(A + A^T) = \begin{bmatrix} 1 & 4 & \frac{7}{2} \\ 4 & 8 & 3 \\ \frac{7}{2} & 3 & 7 \end{bmatrix}\) – symmetric.
- \(Q = \tfrac12(A - A^T) = \begin{bmatrix} 0 & -2 & \frac{1}{2} \\ 2 & 0 & -2 \\ - \frac{1}{2} & 2 & 0 \end{bmatrix}\) – skew-symmetric (zero diagonal).
- \(P + Q = A\). ✓
Final Answer: \(A = P + Q\) as shown
Inverse of a 2 × 2 matrix
Matrices · Invertible MatricesFind the inverse of \(A = \begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}\) and verify.
- \(ad - bc = 15 - 14 = 1 \ne 0\).
- \(A^{-1} = \begin{bmatrix} 5 & -2 \\ -7 & 3 \end{bmatrix}\).
- Check: \(AA^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\). ✓
Final Answer: \(A^{-1} = \begin{bmatrix} 5 & -2 \\ -7 & 3 \end{bmatrix}\)
Practice & Tests
Chapter Practice
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