Proportionality Property in Determinants

The proportionality property of determinants is a fundamental concept in linear algebra, particularly when dealing with the determinant of a square matrix. This property has implications for understanding when a system of linear equations has a unique solution, and it is also used in various applications such as physics, engineering, and computer science.

Understanding the Proportionality Property

The proportionality property states that if one row (or column) of a square matrix is multiplied by a scalar, the determinant of the matrix is also multiplied by that scalar.

Mathematically, if $A$ is a square matrix and $k$ is a scalar, then:

$$ \text{det}(k \cdot A_i) = k \cdot \text{det}(A) $$

where $A_i$ represents the $i$-th row or column of the matrix $A$ after being multiplied by $k$, and $\text{det}(A)$ is the determinant of the original matrix $A$.

Formulas Involving Proportionality Property

Let's consider a square matrix $A$ of order $n \times n$:

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} $$

If we multiply the $i$-th row by a scalar $k$, the new matrix $A'$ is:

$$ A' = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ \vdots & \vdots & \ddots & \vdots \ k \cdot a_{i1} & k \cdot a_{i2} & \cdots & k \cdot a_{in} \ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} $$

The determinant of $A'$ is:

$$ \text{det}(A') = k \cdot \text{det}(A) $$

Examples

Example 1: Scalar Multiplication of a Row

Consider the matrix:

$$ B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} $$

The determinant of $B$ is:

$$ \text{det}(B) = (1)(4) - (2)(3) = 4 - 6 = -2 $$

Now, let's multiply the first row by a scalar $k = 3$:

$$ B' = \begin{bmatrix} 3 \cdot 1 & 3 \cdot 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \ 3 & 4 \end{bmatrix} $$

The determinant of $B'$ is:

$$ \text{det}(B') = (3)(4) - (6)(3) = 12 - 18 = -6 $$

As expected, $\text{det}(B') = 3 \cdot \text{det}(B)$.

Example 2: Scalar Multiplication of a Column

Consider the same matrix $B$:

$$ B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} $$

Now, let's multiply the second column by a scalar $k = 5$:

$$ B'' = \begin{bmatrix} 1 & 5 \cdot 2 \ 3 & 5 \cdot 4 \end{bmatrix} = \begin{bmatrix} 1 & 10 \ 3 & 20 \end{bmatrix} $$

The determinant of $B''$ is:

$$ \text{det}(B'') = (1)(20) - (10)(3) = 20 - 30 = -10 $$

Again, $\text{det}(B'') = 5 \cdot \text{det}(B)$.

Table of Differences and Important Points

Property	Description	Implication
Linearity	If two rows (or columns) are proportional, the determinant is zero.	Useful for checking linear dependence.
Proportionality	Multiplying a row (or column) by a scalar multiplies the determinant by the same scalar.	Allows for scaling of the determinant.

Conclusion

The proportionality property of determinants is a powerful tool in linear algebra. It simplifies the computation of determinants after row or column operations and provides insights into the behavior of linear transformations represented by matrices. Understanding this property is essential for students and professionals who work with systems of linear equations and matrix algebra.