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A sparse $4 \times 4$ determinant is $30$ by both cofactor expansion and row reduction

Example

Over $Z$ ,

$A = (\begin{array}{cccc} 1 & 2 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 4 & 8 & 2 & 1 \\ 0 & 0 & 0 & 5 \end{array})$

has determinant $30$ , both by cofactor expansion and by row reduction.

Facts & Assumptions

Given: The displayed matrix $A$ .

[F1]

$Z$ is a commutative ring (The integers form a commutative ring).

[L1]

Laplace expansion computes a determinant along every row and column (Laplace expansion computes the determinant along every row and every column over a commutative ring).

[L2]

Adding a multiple of one row to another leaves the determinant unchanged (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).

[L3]

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

Verification

technique · direct

1.1

Expand along the last row. Its only nonzero entry is $5$ in position $(3, 3)$ , with positive cofactor sign, so $\det (A) = 5 \det (\begin{array}{ccc} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 4 & 8 & 2 \end{array}) .$

F1L1algebra

1.2

Alternatively, replace row $2$ by row $2 - 4$ times row $0$ (the third row by the third row minus four times the first row). The result is upper triangular with diagonal $1, 3, 2, 5$ , and the operation does not change the determinant.

L2algebra

2.1

Expand the $3 \times 3$ determinant along its last column. The only nonzero entry there is $2$ , again with positive sign, giving $\det (A) = 5 \cdot 2 \det (\begin{array}{cc} 1 & 2 \\ 0 & 3 \end{array}) = 5 \cdot 2 \cdot 3 = 30$ .

step 1.1L1algebra

2.2

By [L3], the determinant in step 1.2 is $1 \cdot 3 \cdot 2 \cdot 5 = 30$ .

step 1.2L3algebra

3.1

Steps 2.1 and 2.2 give the same value by the two requested methods.

step 2.1step 2.2∎

Depends on

Used by

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A sparse 4×44×4 determinant is 3030 by both cofactor expansion and row reduction