Factor Property in Determinants

The factor property of determinants is an important concept in linear algebra, particularly when dealing with determinants of matrices. This property allows us to factor out a common scalar from a row or a column of a matrix.

Understanding the Factor Property

The factor property states that if all elements of a row or a column of a square matrix are multiplied by a scalar, the determinant of the matrix is multiplied by the same scalar.

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

$$ \text{If } A' \text{ is obtained by multiplying a row (or column) of } A \text{ by } k, \text{ then } \det(A') = k \cdot \det(A) $$

This can be represented for a $3 \times 3$ matrix as follows:

$$ \det\begin{pmatrix} a & b & c \ d & e & f \ g & h & i \ \end{pmatrix} = k \cdot \det\begin{pmatrix} a & b & c \ kd & ke & kf \ g & h & i \ \end{pmatrix} $$

Here, the second row of the matrix has been multiplied by the scalar $k$.

Table of Differences and Important Points

Property	Description	Example
Factor Property	Allows factoring out a scalar from a row or column	$\det(k \cdot A) = k^n \cdot \det(A)$ for an $n \times n$ matrix
Linearity	Determinant is a linear function with respect to each row/column	$\det(A + B) \neq \det(A) + \det(B)$
Multiplicative Property	The determinant of the product of matrices equals the product of their determinants	$\det(AB) = \det(A) \cdot \det(B)$

Formulas Involving the Factor Property

The factor property can be generalized for an $n \times n$ matrix. If a matrix $A$ has a row or a column where each element is multiplied by a scalar $k$, then the determinant of the new matrix is $k$ times the determinant of the original matrix.

$$ \det(k \cdot A) = k^n \cdot \det(A) $$

Here, $n$ is the size of the matrix, and the scalar $k$ is raised to the power of $n$ because it is multiplied across $n$ elements of the row or column.

Examples to Explain Important Points

Example 1: Factoring from a Single Row

Consider the $3 \times 3$ matrix $A$:

$$ A = \begin{pmatrix} 2 & 4 & 6 \ 1 & 3 & 5 \ 7 & 8 & 9 \ \end{pmatrix} $$

If we multiply the second row by $3$, the new matrix $A'$ is:

$$ A' = \begin{pmatrix} 2 & 4 & 6 \ 3 & 9 & 15 \ 7 & 8 & 9 \ \end{pmatrix} $$

Using the factor property, the determinant of $A'$ is:

$$ \det(A') = 3 \cdot \det(A) $$

Example 2: Factoring from Multiple Rows

If we multiply each row of a $2 \times 2$ matrix by different scalars, the determinant changes as follows:

$$ B = \begin{pmatrix} a & b \ c & d \ \end{pmatrix} $$

Multiplying the first row by $k$ and the second row by $l$, we get:

$$ B' = \begin{pmatrix} ka & kb \ lc & ld \ \end{pmatrix} $$

The determinant of $B'$ is:

$$ \det(B') = kl \cdot \det(B) $$

Example 3: Factoring from a Column

Similarly, if we have a matrix $C$ and we multiply a column by a scalar $m$, the determinant of the new matrix $C'$ is $m$ times the determinant of $C$.

$$ C = \begin{pmatrix} x & y \ z & w \ \end{pmatrix} $$

Multiplying the first column by $m$, we get:

$$ C' = \begin{pmatrix} mx & y \ mz & w \ \end{pmatrix} $$

The determinant of $C'$ is:

$$ \det(C') = m \cdot \det(C) $$

Conclusion

The factor property is a powerful tool in the computation and manipulation of determinants. It simplifies the process of evaluating determinants, especially when dealing with matrices that have common factors in their rows or columns. Understanding this property is essential for solving problems in linear algebra and related fields.