AcademyDeterminants
Academy
Determinant Properties
Level 1 - Linear Algebra topic page in Determinants.
Principle
Determinant properties explain how the determinant changes under row operations and matrix operations. These rules are usually faster and more informative than expanding from the definition.
The key idea is that the determinant is linear in each row or column and changes sign when two rows or columns are swapped.
Notation
\(A,B\)
square matrices of the same size
\(R_i\)
row i
\(\lambda\)
a scalar
\(A^T\)
transpose of A
\(I_n\)
n by n identity matrix
The Core Method
Use these determinant rules:
Identity determinant
\[\det I_n=1\]
Transpose determinant
\[\det(A^T)=\det A\]
Product determinant
\[\det(AB)=\det A\det B\]
Row operation effects:
- Swapping two rows multiplies the determinant by \(-1\).
- Multiplying one row by \(\lambda\) multiplies the determinant by \(\lambda\).
- Adding a multiple of one row to another row does not change the determinant.
If a matrix has a zero row, repeated row, or one row that is a scalar multiple of another row, its determinant is zero.
Worked Cases
Question
A matrix \(A\) is transformed into \(B\) by swapping two rows, then multiplying one row by \(5\). If \(\det B=20\), find \(\det A\).
Answer
A row swap multiplies the determinant by \(-1\). Multiplying one row by \(5\) then multiplies the determinant by \(5\). Therefore \(\det B=-5\det A\). Since \(20=-5\det A\), divide by \(-5\) to get \(\det A=-4\).
Examples
Question
Why does a matrix with two equal rows have determinant zero?
Answer
Swap the two equal rows. A row swap should multiply the determinant by \(-1\). But the matrix is unchanged because the rows are equal, so the determinant is also unchanged. Therefore \(\det A=-\det A\), giving \(2\det A=0\), so \(\det A=0\).