Problem 11 - Precalculus: Functions and Graphs

Question

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{array}\right|=0$$

Step-by-Step Solution

Answer

The determinant is zero because the matrix has a column filled entirely with zeros.

Step 1: Understanding Determinant Properties

A determinant of a matrix with all zero elements in any row or column is zero. This is because multiplying by zero in the cofactor expansion will result in a sum of zero, as each term will have at least one factor of zero.

Step 2: Identifying the Zero Row/Column

We need to inspect the matrix to identify a row or column filled with zeros. In the given matrix, the second column is completely filled with zeros: \(\begin{array}{lll} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 0 & 2 \end{array}\).

Step 3: Applying Determinant Property

Since the second column has all zero elements, using the property of determinants, the determinant of the matrix is zero regardless of the other elements filled in the matrix. Thus, \(\left|\begin{array}{lll} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 0 & 2 \end{array}\right| = 0\).

Key Concepts

Zero Rows and Columns in Determinants

In any matrix, if an entire row or column is composed solely of zeroes, the determinant of that matrix is zero. This is because when you calculate a determinant, you use a method called cofactor expansion. During this process, if you expand along a row or column containing only zeroes, each term in your calculation will include a zero somewhere in it.
Consequently, when you sum all these terms, the total will also be zero. Therefore, when you see a matrix, like in the original problem, where a whole column or row is filled with zeroes, you can immediately conclude its determinant is zero without needing to perform laborious calculations.
For example, in the matrix: \[\begin{array}{lll} 0 & 0 & 1 \1 & 0 & 0 \0 & 0 & 2 \end{array}\]the second column is all zero, making the determinant instantly zero.

Matrix Properties

Matrices have unique properties that help us perform operations efficiently. One of these properties is related to determinants and zero rows or columns. Understanding these properties allows us to handle complex matrices by breaking them down into simpler components.
Besides zero rows or columns, here are some key matrix properties important in determinants:

• If the matrix has two identical rows or columns, its determinant is zero.
• Swapping two rows or columns changes the sign of the determinant.
• Adding a multiple of one row to another does not change its determinant.

Recognizing these properties can save time and effort and help you understand when and why a determinant equals zero simply by inspecting the matrix.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method used to calculate the determinant of a matrix. Here’s a quick breakdown of how it works:

• Choose any row or column to perform the expansion. Some mathematicians recommend picking the row or column with the most zeroes to simplify calculations.
• For each element in the row or column, compute its cofactor. A cofactor is the signed minor of the element. The sign depends on the element's position, determined by \( (-1)^{i+j} \), where \( i \) and \( j \) are the row and column numbers, respectively.
• The determinant is the sum of products of each element and its corresponding cofactor.

If a row or column is filled with zeroes, as in our example, all terms in this sum will contain a zero, and therefore the sum will be zero, confirming that the determinant is zero.
This method not only helps in finding determinants but also in understanding the structure of matrices.