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Properties of Determinants

Determinant-properties demo. Enter A to numerically illustrate transposition invariance, row-swap sign flip and proportional-row zero—for intuitive learning.

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Row swap flips sign; rᵢ→k·rᵢ scales det by k; det(Aᵀ)=det(A)
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Properties of Determinants

1. Definition
Three useful properties: row swap flips sign, scaling a row scales det, transpose does not change det.
2. Symbols
SymbolMeaning
det(A)original determinant
−det(A)after row swap
k·det(A)after row scaling
3. How this tool works
  • Swapping two rows (columns) flips the sign;
  • Multiplying a row (column) by k multiplies det by k;
  • Transposing does not change det.
4. Steps
  1. Enter A;
  2. Click Calculate to see numerical verification of the three properties.
5. Example
Example: A=1 2;3 4, det=−2. After row swap det=2=−(−2); after scaling row 1 by 3 det=−6=3·(−2).
6. Pitfalls
Scaling one row by k multiplies only that row, not the whole matrix;
A row swap flips the sign only;
Transposing leaves det unchanged.

❓ FAQ｜Properties of Determinants

What happens after a row swap?
The sign flips, the absolute value stays.
What if a row is multiplied by k?
The determinant is multiplied by k.
What is det(Aᵀ)?
It equals det(A).
What is det(kA)?
For an n×n matrix, kⁿ·det(A).
What is det(AB)?
det(A)·det(B).