All Zero Property in Determinants

The determinant of a matrix is a special number that can be calculated from its elements and has many applications in linear algebra, geometry, and differential equations. One of the properties of determinants is the "All Zero Property," which is related to matrices that have rows or columns with all zero elements.

Understanding the All Zero Property

The All Zero Property states that if any row or column of a square matrix is composed entirely of zeros, then the determinant of that matrix is zero. This property is a direct consequence of the definition of the determinant and the way it is calculated.

Mathematical Representation

For a square matrix $A$ of order $n \times n$, if the $i$-th row or $j$-th column consists entirely of zeros, then:

$$ \text{det}(A) = 0 $$

This can be represented as:

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \quad \text{or} \quad A = \begin{bmatrix} a_{11} & \cdots & 0 & \cdots & a_{1n} \ a_{21} & \cdots & 0 & \cdots & a_{2n} \ \vdots & \ddots & \vdots & \ddots & \vdots \ a_{n1} & \cdots & 0 & \cdots & a_{nn} \end{bmatrix} $$

In both cases, $\text{det}(A) = 0$.

Why the All Zero Property Holds

The determinant of a matrix can be thought of as a sum of products of elements from each row or column, with each product multiplied by a sign that depends on the position of the elements. When a row or column is all zeros, every product that includes an element from that row or column will be zero. Since the determinant is the sum of these products, if all of them are zero, the determinant must also be zero.

Table of Differences and Important Points

Property	Description	Consequence
Non-Zero Rows/Columns	No row or column is entirely zero	Determinant may be non-zero
All Zero Row/Column	At least one row or column is entirely zero	Determinant is zero

Examples

Example 1: All Zero Row

Consider the following $3 \times 3$ matrix with an all-zero third row:

$$ A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 0 & 0 & 0 \end{bmatrix} $$

The determinant of $A$ is:

$$ \text{det}(A) = 0 \cdot \text{det}\left(\begin{bmatrix}2 & 3\5 & 6\end{bmatrix}\right) - 0 \cdot \text{det}\left(\begin{bmatrix}1 & 3\4 & 6\end{bmatrix}\right) + 0 \cdot \text{det}\left(\begin{bmatrix}1 & 2\4 & 5\end{bmatrix}\right) = 0 $$

Example 2: All Zero Column

Consider the following $3 \times 3$ matrix with an all-zero second column:

$$ B = \begin{bmatrix} 1 & 0 & 3 \ 4 & 0 & 6 \ 7 & 0 & 9 \end{bmatrix} $$

The determinant of $B$ is:

$$ \text{det}(B) = 1 \cdot \text{det}\left(\begin{bmatrix}0 & 6\0 & 9\end{bmatrix}\right) - 0 \cdot \text{det}\left(\begin{bmatrix}4 & 6\7 & 9\end{bmatrix}\right) + 3 \cdot \text{det}\left(\begin{bmatrix}4 & 0\7 & 0\end{bmatrix}\right) = 0 $$

In both examples, the presence of an all-zero row or column leads to a determinant of zero, illustrating the All Zero Property.

Conclusion

The All Zero Property is a fundamental aspect of determinants that simplifies the calculation and analysis of matrices. It is particularly useful when determining if a matrix is singular (non-invertible), as a zero determinant indicates that the matrix does not have an inverse. Understanding this property is essential for students and professionals working with linear algebra and related fields.