Determinant of scalar product of matrix

Dependencies:

1. Determinant after elementary row operation

Let $c$ be a scalar and $A$ be an $n$ by $n$ matrix. Then $|cA| = c^n|A|$.

Proof

$cA$ can be obtained from $A$ by applying the elementary row operation $\langle i \rangle \leftarrow c\langle i \rangle$ on each row.

Therefore, $|cA| = c(c(\ldots(c(c|A|))\ldots)) = c^n|A|$.

Dependency for: None

Info:

• Depth: 7
• Number of transitive dependencies: 9

Transitive dependencies:

1. Group
2. Ring
3. Semiring
4. Matrix
5. Submatrix
6. Determinant
7. Swapping last 2 rows of a matrix negates its determinant
8. Elementary row operation
9. Determinant after elementary row operation