Problem 8 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The determinant is zero because the first and third rows are linearly dependent.

Step 1: Understand the Matrix

Examine the given 3x3 matrix: \[ \begin{bmatrix} 1 & -1 & 1 \ 0 & 1 & 0 \ -1 & 1 & -1 \end{bmatrix} \]. Notice that the first row is \([1, -1, 1]\), the second row is \([0, 1, 0]\), and the third row is \([-1, 1, -1]\).

Step 2: Identify Linear Dependence

Linear dependence occurs when one row (or column) of a matrix can be written as a linear combination of other rows (or columns). Upon examination, the third row \([-1, 1, -1]\) is the negative of the first row \([1, -1, 1]\), implying these two rows are linearly dependent.

Step 3: Establish the Determinant Condition

One property of determinants is that if any two rows of a matrix are linearly dependent, then the determinant of that matrix is zero. In this case, because the first and third rows are linearly dependent, the determinant must be zero.

Key Concepts

Linear Dependence

Linear dependence is a situation within a matrix where at least one row or column can be exactly described as a linear combination of the others. This means that one row or column can be obtained by adding, subtracting, or scaling other rows or columns. In our specific example, the 3x3 matrix has its third row \([-1, 1, -1]\), which interestingly stands as the negative of the first row \([1, -1, 1]\). This creates a scenario of linear dependence because the third row can directly be obtained by multiplying the first row by -1.

• First row: \([1, -1, 1]\)
• Third row: multiply first row by -1 to get \([-1, 1, -1]\)

Thus, any matrix where rows or columns can be expressed this way cannot retain full rank. Always remember: a linear combination within rows or columns signals linear dependence.

Property of Determinants

The determinant of a matrix is a unique number associated with the matrix itself. It holds several important properties, one of which is directly linked to linear dependence. Specifically, if a matrix has any two rows or columns that are linearly dependent, the determinant of that matrix becomes zero. This property tells us something essential about the structure of the matrix, often relating to geometry like volume or area in higher dimensions.

• Determinant becomes zero if there's linear dependence
• Reflects the 'singularity' of the matrix (no unique solutions)
• Helps in identifying invertible matrices (only non-zero determinants)

In the problem we've tackled, since the first and third rows of our matrix are linearly dependent, we immediately conclude that the determinant must be zero. This is an integral property of determinants that helps simplify checking for singular matrices or matrix invertibility.

Matrix Row Operations

Matrix row operations are crucial when analyzing or simplifying a matrix, especially in solving equations or calculating determinants. Row operations like swapping, multiplying, or adding rows are used to alter the matrix without changing its fundamental properties. But here's a critical piece: these operations can reveal characteristics like linear dependence. When analyzing, we found by inspection that row 3 is just row 1 scaled by -1 (an operation itself), which explained the linear dependence directly. In other circumstances, row operations can help rework the matrix into simpler forms, like the Identity Matrix, where the determinant becomes more apparent.

• Swap rows: changes sign of determinant
• Multiply a row: scales the determinant
• Add/Subtract rows: helpful for gauging dependence

In our task, direct comparison substituted the need for further operations, emphasizing the importance of recognizing simple relations and dependencies at a glance. Row operations can be powerful, but knowing when and how to employ them is key.