False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: substituting a fixed matrix $A$ for $x$ defines a ring homomorphism $M_{n} (F [x]) \to M_{n} (F)$

Statement refuted

Refuted claim: For a fixed $A \in M_{n} (F)$ , the coefficientwise rule $\sum_{k} C_{k} x^{k} \mapsto \sum_{k} C_{k} A^{k}$ is a ring homomorphism $M_{n} (F [x]) \to M_{n} (F)$ .

This is the invalid substitution step in a familiar pseudo-proof of Cayley-Hamilton; the actual theorem Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, $χ_{T} (T) = 0$ requires a coefficient-comparison argument.

Facts & Assumptions

Given: Work over any field $F$ and take $A = diag (1, 0)$ and $B = E_{12}$ in $M_{2} (F)$ .

[L1]

Matrix multiplication over a commutative ring is defined by finite row-column sums, and the identity matrix has $1$ on its diagonal and $0$ elsewhere (Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose).

Refutation

technique · explicit counterexample

1.1

In $M_{2} (F [x])$ , the row-column formula [L1] and commutativity of $F [x]$ give $(x I_{2}) B = B (x I_{2}) = x B$ .

L1algebra

2.1

The coefficientwise rule has $E (x I_{2}) = A$ , $E (B) = B$ , and $E ((x I_{2}) B) = E (x B) = B A = 0$ , whereas $E (x I_{2}) E (B) = A B = B$ .

step 1.1givenalgebra

3.1

Since $B \neq 0$ , step 2.1 shows $E ((x I_{2}) B) \neq E (x I_{2}) E (B)$ . The coefficientwise substitution rule is therefore not multiplicative and hence is not a ring homomorphism.

step 2.1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 41 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

P. Garrett, Cayley-Hamilton notes (standard reference, not scraped)
H. Haber, Notes on the characteristic polynomial (standard reference, not scraped)

FALSE: substituting a fixed matrix AA for xx defines a ring homomorphism Mn(F[x])→Mn(F)Mn​(F[x])→Mn​(F)