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Row operations track determinant correctly for a singular triangular matrix over $\mathbb Z/6$

Example

Over $R=\mathbb Z/6$ , let $A=\operatorname{diag}(2,1)$ . Then $A$ is not invertible although $\det A=2\ne0$ . A swap changes the determinant to $4=-2$ , scaling the first row by the nonzero nonunit $3$ changes it to $0=3\cdot2$ , and adding four times the second row to the first leaves it equal to $2$ .

Facts & Assumptions

Given: The displayed matrix over $\mathbb Z/6$ .

[L1]

Row swaps, arbitrary scalings and row additions have the three determinant laws without a nonsingularity assumption (For every square matrix, including singular ones, a row swap negates the determinant, scaling a row by any scalar scales it, and row addition leaves it unchanged).

[L2]

A triangular determinant is the product of its diagonal entries (The determinant of a triangular matrix is the product of its diagonal entries).

[L3]

Arithmetic in $\mathbb Z/6$ is commutative ring arithmetic modulo $6$ (For every natural $n$ , $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

[L4]

$\mathbb Z/6$ consists of congruence classes modulo $6$ (The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$ ).

[L5]

An invertible matrix over a commutative ring has unit determinant (An invertible square matrix over a commutative ring has unit determinant).

Verification

technique · direct

1.1

By [L2], $\det A=2$ . The element $2$ is not a unit modulo $6$ , since every product $2r$ is even and cannot be congruent to $1$ ; therefore [L5] implies that $A$ is not invertible.

L1L2L3L4L5algebra

2.1

Swapping rows gives $\begin{pmatrix}0&1\\2&0\end{pmatrix}$ with determinant $-2=4$ . Scaling row $0$ by $3$ gives $\operatorname{diag}(0,1)$ with determinant $0=3\cdot2$ . Replacing row $0$ by row $0+4R_1$ gives $\begin{pmatrix}2&4\\0&1\end{pmatrix}$ with determinant $2$ .

step 1.1algebra

3.1

These values agree respectively with negation, multiplication by $3$ , and invariance under row addition in [L1]. No cancellation, unit scaling or invertibility was used.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 83 results over 23 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

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4 (standard reference, not scraped)

Row operations track determinant correctly for a singular triangular matrix over Z/6\mathbb Z/6Z/6