Testing Universal Claims with Small Matrices

What a Single Counterexample Can Establish

Matrix statements often make claims about every matrix of a specified size: for example, that all pairs of square matrices commute, or that a matrix with square zero must itself be zero. A proof of such a statement must cover every matrix in its stated domain. A disproof has a lighter burden: one matrix, or one pair of matrices, satisfying the claim’s hypotheses but violating its conclusion is enough. The practical challenge is to find a small witness and verify it without making a computational error.

Throughout this tutorial, matrices have real entries. For a positive integer \(n\), write \(\mathbb{R}^{n\times n}\) for the set of \(n\times n\) matrices, \(\mathbf{O}\) for the zero matrix of the required size, and \(\mathbf{I}_n\) for the \(n\times n\) identity matrix. Matrix equality means equality of corresponding entries. For two \(2\times2\) matrices, the product is computed by taking row-by-column products; for instance, the entry in row \(i\), column \(j\) of \(\mathbf{A}\mathbf{B}\) is \(a_{i1}b_{1j}+a_{i2}b_{2j}\).

The domain and the quantifiers matter. A \(2\times2\) example refutes a claim about all square matrices, since \(2\times2\) matrices are among the square matrices. It does not by itself refute a claim specifically about \(3\times3\) matrices. Likewise, to refute a statement about every pair \((\mathbf{A},\mathbf{B})\), provide one pair; to refute a statement about every individual \(\mathbf{A}\), provide one matrix. First match the witness to the domain, then check the property.

Proof. Suppose the universal claim is false. Then it is not true that every member of \(S\) has property \(P\). Therefore at least one member \(\mathbf{A}_0\in S\) does not have property \(P\), which is the required witness. Conversely, suppose there is a matrix \(\mathbf{A}_0\in S\) for which \(P(\mathbf{A}_0)\) is false. A statement that \(P\) holds for every member of \(S\) would in particular require it to hold for \(\mathbf{A}_0\). Since it does not, the universal claim is false. \(\square\)

For a conditional claim, the witness must satisfy its hypothesis. A matrix that fails the conclusion but also fails the hypothesis does not refute the implication.

Proof. By the definition of an implication, the statement “if \(P(\mathbf{A})\), then \(Q(\mathbf{A})\)” is false exactly when its hypothesis \(P(\mathbf{A})\) is true and its conclusion \(Q(\mathbf{A})\) is false. Thus, if the universal implication fails, it fails for at least one matrix \(\mathbf{A}_0\in S\), and that matrix has precisely those two properties. Conversely, if such a matrix \(\mathbf{A}_0\) exists, the implication fails for \(\mathbf{A}_0\), so it cannot hold for every matrix in \(S\). \(\square\)

Choose Matrices That Expose the Claim

Before trying random entries, look for matrices whose structure makes the relevant calculation short. The zero matrix can test claims involving products or sums equaling zero. The identity matrix often simplifies products, because multiplying by \(\mathbf{I}_n\) leaves a matrix unchanged. A diagonal matrix has zero entries outside its main diagonal, so many products and powers are easy to calculate. A nilpotent matrix is a square matrix \(\mathbf{N}\) for which \(\mathbf{N}^k=\mathbf{O}\) for some positive integer \(k\); its powers can reveal why a claim about a zero power is too strong.

These are useful starting points, not automatic counterexamples. A candidate still has to belong to the stated domain, satisfy any hypotheses, and actually violate the conclusion. The following examples show how those checks work.

How to Make a Counterexample Check Complete

A reliable counterexample has three parts: the candidate, the computation establishing the hypothesis, and the comparison showing the conclusion fails. For a matrix equality, the last part can be a single differing entry, but the products or powers that lead to that entry must be computed correctly. For a conditional claim, state explicitly which hypothesis holds and which conclusion does not. This makes the logical role of the calculation visible instead of leaving the reader to infer it.

Testing a few simple matrices is not a substitute for a proof of a universal claim. If every matrix you try satisfies the statement, that only tells you that those particular tests did not refute it. Conversely, once a valid counterexample is found, further examples are unnecessary to establish that the universal claim is false.

Common Pitfall: Reversing the Product

Matrix multiplication depends on the order of the factors, so a row-by-column error can give the wrong intermediate answer. With the matrices in the first example, a tempting but incorrect calculation is to say that \(\mathbf{A}\mathbf{B}=\mathbf{A}\) and \(\mathbf{B}\mathbf{A}=\mathbf{O}\). That has the products reversed. The correction is to keep the written order: computing rows of the left factor against columns of the right factor gives \(\mathbf{A}\mathbf{B}=\mathbf{O}\) and \(\mathbf{B}\mathbf{A}=\mathbf{A}\). The pair still disproves commutativity, but an accidentally reversed calculation is not a sound verification. Write the factors above the computation and check each entry using the product definition.

A related logical error is to choose a matrix that violates the conclusion but not the hypothesis. Such a matrix cannot refute an implication. For instance, to challenge “if \(\mathbf{A}^2=\mathbf{O}\), then \(\mathbf{A}=\mathbf{O}\),” it is not enough to choose any nonzero matrix: its square must also be checked. The nilpotent matrix above works because the calculation verifies both required facts.

Zero, identity, diagonal, and nilpotent matrices are useful because their entries make the verification transparent. Their value is not that they settle every question in advance, but that they make good first candidates: each can expose a specific overgeneralization while keeping the arithmetic small enough to audit.