Property of invariance
Property of Invariance in Determinants
The property of invariance in the context of determinants refers to certain transformations that can be applied to the rows or columns of a square matrix without changing the value of its determinant. Understanding these properties is crucial for solving problems in linear algebra, particularly those involving determinants.
Basic Definition
A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation represented by the matrix.
Properties of Determinants
Here are some key properties of determinants that exhibit invariance:
Multiplying a Row or Column by a Scalar: If a row (or column) of a matrix is multiplied by a scalar, the determinant of the matrix is multiplied by the same scalar.
Swapping Rows or Columns: Swapping two rows (or columns) of a matrix multiplies the determinant by -1.
Adding a Multiple of a Row or Column to Another: Adding a multiple of one row (or column) to another row (or column) does not change the determinant.
Row or Column of Zeros: If a matrix has a row or column of all zeros, its determinant is zero.
Duplicate Rows or Columns: If a matrix has two identical rows (or columns), its determinant is zero.
Upper or Lower Triangular Matrix: The determinant of an upper or lower triangular matrix is the product of its diagonal elements.
Table of Invariance Properties
| Property | Description | Effect on Determinant |
|---|---|---|
| Scalar Multiplication | Multiply a row/column by scalar k |
Determinant is multiplied by k |
| Row/Column Swap | Swap two rows/columns | Determinant is multiplied by -1 |
| Row/Column Addition | Add a multiple of a row/column to another | Determinant is unchanged |
| Zero Row/Column | A row/column of all zeros | Determinant is 0 |
| Duplicate Rows/Columns | Two identical rows/columns | Determinant is 0 |
| Triangular Matrix | Matrix is upper/lower triangular | Determinant is product of diagonal elements |
Formulas
The determinant of a matrix $A$ is denoted by $\det(A)$ or $|A|$. For a $2 \times 2$ matrix:
$$ \det(A) = \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc $$
For a $3 \times 3$ matrix:
$$ \det(A) = \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = aei + bfg + cdh - ceg - bdi - afh $$
Examples
Example 1: Scalar Multiplication
Consider the matrix $A$:
$$ A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} $$
The determinant of $A$ is:
$$ \det(A) = (1)(4) - (2)(3) = 4 - 6 = -2 $$
If we multiply the first row by a scalar $k = 3$, the new matrix $A'$ is:
$$ A' = \begin{pmatrix} 3 & 6 \ 3 & 4 \end{pmatrix} $$
The determinant of $A'$ is:
$$ \det(A') = (3)(4) - (6)(3) = 12 - 18 = -6 $$
Notice that $\det(A') = 3 \cdot \det(A)$, which confirms the property of scalar multiplication.
Example 2: Row Addition
Consider the matrix $B$:
$$ B = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} $$
The determinant of $B$ is:
$$ \det(B) = (1)(4) - (2)(3) = 4 - 6 = -2 $$
If we add the first row to the second row, the new matrix $B'$ is:
$$ B' = \begin{pmatrix} 1 & 2 \ 4 & 6 \end{pmatrix} $$
The determinant of $B'$ is:
$$ \det(B') = (1)(6) - (2)(4) = 6 - 8 = -2 $$
The determinant remains unchanged, illustrating the property of row addition.
Example 3: Triangular Matrix
Consider the upper triangular matrix $C$:
$$ C = \begin{pmatrix} 2 & 3 \ 0 & 5 \end{pmatrix} $$
The determinant of $C$ is simply the product of its diagonal elements:
$$ \det(C) = (2)(5) = 10 $$
This example demonstrates the property of determinants for triangular matrices.
Understanding these properties is essential for simplifying the computation of determinants and for proving various results in linear algebra. They are also widely used in solving systems of linear equations, finding eigenvalues, and in many applications across physics, engineering, and computer science.