1. Why Properties of Determinants Matter
The properties of determinants help us simplify calculations, especially for large matrices. Instead of expanding the determinant directly, we can use these rules to make the matrix easier to work with. These notes explain each property in simple language with examples.
2. Property 1: Interchanging Two Rows or Two Columns
If you swap (interchange) any two rows or any two columns of a matrix, the sign of the determinant changes.
R_i \leftrightarrow R_j \Rightarrow |A| \rightarrow -|A|
2.1. Example
Original matrix:
A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad |A| = -2
After swapping the two rows:
A' = \begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix}, \quad |A'| = +2
The sign changed as expected.
3. Property 2: If Two Rows or Two Columns Are Identical
If any two rows or any two columns of a matrix are the same, the determinant is zero.
This is because identical rows/columns make the matrix linearly dependent.
4. Property 3: If a Row or Column Is All Zeros
If any row or column consists entirely of zeros, the determinant is zero.
No calculation is needed; the determinant collapses automatically.
5. Property 4: Multiplying a Row or Column by a Scalar
If you multiply a row or column by a scalar \( k \), the determinant also gets multiplied by \( k \).
R_i \rightarrow kR_i \Rightarrow |A| \rightarrow k|A|
5.1. Example
Let:
|A| = 5
If we multiply a row by 3, the new determinant becomes:
|A'| = 3 \times 5 = 15
6. Property 5: Adding a Multiple of One Row to Another Row
If you add a multiple of one row to another row (or one column to another), the determinant does not change.
R_i \rightarrow R_i + kR_j \Rightarrow |A'| = |A|
This is one of the most useful properties for simplifying matrices before expanding the determinant.
7. Property 6: Determinant of a Triangular Matrix
If a matrix is upper triangular, lower triangular, or diagonal, its determinant is simply the product of its diagonal elements.
|A| = a_{11} a_{22} a_{33} \cdots a_{nn}
7.1. Example
For the matrix:
A = \begin{bmatrix} 2 & -1 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}
Determinant:
|A| = 2 \times 4 \times 6 = 48
8. Property 7: Determinant of the Product of Two Matrices
If \( A \) and \( B \) are two square matrices of the same order, then:
|AB| = |A| \cdot |B|
This is extremely helpful when working with invertible matrices.
9. Property 8: Transpose and Determinant
The determinant of a matrix and its transpose are equal.
|A^T| = |A|
10. Quick Recap of All Properties
- Swapping rows/columns → sign changes
- Two identical rows/columns → determinant = 0
- A zero row/column → determinant = 0
- Scaling a row/column → scaling determinant
- Row replacement → determinant unchanged
- Triangular matrix → product of diagonals
- det(AB) = det(A)det(B)
- det(A) = det(Aᵀ)
These rules make working with determinants faster and easier.