Sum Property in Determinants

The sum property in the context of determinants is an important concept in linear algebra. It deals with how the determinant of a matrix changes when a multiple of one row (or column) is added to another row (or column). This property is crucial for simplifying the calculation of determinants and for understanding the effects of certain row operations on the determinant.

Basic Definition

The determinant of a square matrix $A$ of order $n$ is a scalar value that is a function of the entries of $A$. It is denoted as $\det(A)$ or $|A|$. The determinant has several properties, one of which is the sum property.

Sum Property

The sum property states that if a row (or column) of a determinant is expressed as the sum of two vectors, the determinant can be expressed as the sum of two determinants.

For Rows

If in a determinant, a row is written as the sum of two rows, then the determinant can be split into the sum of two determinants. Mathematically, for a matrix $A$ with rows $R_1, R_2, \ldots, R_n$, if row $R_i$ is expressed as $R_i = R'_i + R''_i$, then:

$$ \begin{vmatrix} R_1 \ \vdots \ R'_i + R''_i \ \vdots \ R_n

\end{vmatrix}

\begin{vmatrix} R_1 \ \vdots \ R'_i \ \vdots \ R_n \end{vmatrix} + \begin{vmatrix} R_1 \ \vdots \ R''_i \ \vdots \ R_n \end{vmatrix} $$

For Columns

Similarly, if in a determinant, a column is written as the sum of two columns, then the determinant can be split into the sum of two determinants. For a matrix $A$ with columns $C_1, C_2, \ldots, C_n$, if column $C_j$ is expressed as $C_j = C'_j + C''_j$, then:

$$ \begin{vmatrix} C_1 & \cdots & C'_j + C''_j & \cdots & C_n

\end{vmatrix}

\begin{vmatrix} C_1 & \cdots & C'_j & \cdots & C_n \end{vmatrix} + \begin{vmatrix} C_1 & \cdots & C''_j & \cdots & C_n \end{vmatrix} $$

Important Points and Differences

Here is a table summarizing the important points and differences related to the sum property:

Aspect	Description
Applicability	Applies to both rows and columns.
Conditions	Only one row or one column can be split at a time.
Effect on Determinant	The determinant is the sum of two determinants.
Linearity	This property is a manifestation of the linearity of the determinant function.
Impact on Computation	Simplifies the computation by breaking down complex determinants into simpler ones.
Preservation of Value	The value of the determinant is preserved; it does not change with the application of this property.

Examples

Let's illustrate the sum property with an example.

Example 1: Row Operation

Consider the determinant of a $3 \times 3$ matrix:

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

Suppose we express the second row as the sum of two vectors: $[4, 5, 6] = [1, 2, 3] + [3, 3, 3]$. Then, by the sum property:

$$ \begin{vmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9

\end{vmatrix}

\begin{vmatrix} 1 & 2 & 3 \ 1 & 2 & 3 \ 7 & 8 & 9 \end{vmatrix} + \begin{vmatrix} 1 & 2 & 3 \ 3 & 3 & 3 \ 7 & 8 & 9 \end{vmatrix} $$

The first determinant on the right-hand side is zero because two rows are identical. The second determinant can be computed normally.

Example 2: Column Operation

Consider the determinant of a $2 \times 2$ matrix:

$$ \begin{vmatrix} a & b+c \ d & e+f \end{vmatrix} $$

By the sum property, we can split the second column:

$$ \begin{vmatrix} a & b+c \ d & e+f

\end{vmatrix}

\begin{vmatrix} a & b \ d & e \end{vmatrix} + \begin{vmatrix} a & c \ d & f \end{vmatrix} $$

Each of these determinants can be computed using the basic formula for a $2 \times 2$ determinant, $ad-bc$.

Conclusion

The sum property is a powerful tool in the computation of determinants. It allows for the simplification of complex determinants and provides insight into the linearity of the determinant function. Understanding this property is essential for solving problems in linear algebra, particularly when dealing with transformations and eigenvalues.