Problem 66 - Algebra and Trigonometry

Question

Matrices with Determinant Zero Use the definition of determinant and the elementary row and column operations to explain why matrices of the following types have determinant 0. (a) A matrix with a row or column consisting entirely of zeros (b) A matrix with two rows the same or two columns the same (c) A matrix in which one row is a multiple of another row, or one column is a multiple of another column

Step-by-Step Solution

Answer

Matrices with a zero row or column, identical rows or columns, or proportional rows or columns all have a determinant of zero due to linear dependence.

Step 1: Understanding the Determinant

The determinant of a matrix is a special value that can be calculated from its elements. It is a scalar that indicates certain properties of the matrix, such as whether it is invertible. A determinant of zero means the matrix is singular and not invertible.

Step 2: Determinant of a Matrix with One Zero Row or Column

A key property of determinants is that if a matrix has a row or a column consisting entirely of zeros, then its determinant is zero. This is because when you expand the determinant using that row or column, all terms will be multiplied by zero, resulting in a determinant of zero.

Step 3: Determinant of a Matrix with Duplicate Rows or Columns

If a matrix has two identical rows or columns, its determinant is zero. This can be shown through the linearity and anti-symmetry property of determinants, where swapping two identical rows or columns changes the sign of the determinant. Swapping identical rows or columns should result in the same determinant, so it must be zero to satisfy both conditions.

Step 4: Determinant of a Matrix with Proportional Rows or Columns

When one row of a matrix is a scalar multiple of another, the determinant is zero. This can be explained by considering elementary row operations: if you add, subtract, or scale rows without changing the determinant, a row that is a multiple of another means the determinant will collapse to zero, analogous to having dependent vectors.

Key Concepts

Elementary Row Operations

Elementary row operations are fundamental tools in linear algebra used to manipulate matrices. These operations include row swapping, scaling a row by a nonzero scalar, and adding a multiple of one row to another. Each operation is pivotal for simplifying matrices and finding solutions to systems of linear equations.

Key characteristics of elementary row operations include:

• Row Swapping: Interchange two rows of a matrix. This operation changes the sign of the matrix's determinant.
• Scaling a Row: Multiply all entries of a row by a nonzero scalar. The determinant is multiplied by this scalar as well, which affects its value.
• Row Addition: Add a multiple of one row to another row. This does not change the determinant.

Each operation is integral for converting a matrix into simpler forms like row echelon form or reduced row echelon form, facilitating easier calculations of determinants. These operations preserve or algebraically manipulate the determinant, helping determine if a matrix is singular, as outlined in the exercise.

Singular Matrix

A singular matrix is one with a determinant of zero, meaning it does not have an inverse. This kind of matrix indicates a loss of information; for example, it may suggest that the equations represented by the matrix are not independent.

Properties of singular matrices include:

• No Unique Solutions: Systems of equations that can be represented by singular matrices often lack unique solutions. They might have infinitely many solutions or none at all.
• Zero Rows/Columns: If a matrix has a full row or column of zeroes, its determinant is zero, making it singular.
• Linearly Dependent Rows/Columns: If any row or column is a linear combination of others, the determinant is zero, leading to a singular matrix.

Understanding singular matrices is crucial since they highlight limitations in data or system constraints, playing a significant role in fields like engineering and computer science.

Dependent Vectors

Dependent vectors in a matrix context mean that some vectors can be expressed as a linear combination of others. In simpler terms, if one vector is a scalar multiple or a sum of others, the set of vectors is not independent.

Implication of dependent vectors in matrices includes:

• Zero Determinant: As seen in exercises involving determinants, a set of dependent vectors ensures the determinant is zero. This is because changing one vector via another does not span more dimensions, reflecting dependency.
• Lack of Unique Solutions: In systems of linear equations, having dependent vectors implies potential redundancy, leading to more than one potential solution or completely undefined solutions.
• No Invertibility: Matrices characterized by dependent vectors become singular, meaning they do not have an inverse. This influences numerical stability and solutions' reliability.

Grasping the concept of dependent vectors is essential for solving and simplifying matrices, as well as understanding their transformations in higher-dimensional spaces.