Question

In Exercises 91 and 92, determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, the determinant will always be zero.

Short answer

The statement is true. If a square matrix has an entire row of zeros, the determinant is indeed zero.

Step-by-step solution

Understanding The Problem

The problem here is stating that if there's a square matrix that has an entire row of zeros, then the determinant will be zero. This statement needs to be justified as true or false based on mathematical knowledge related to determinants.

Justifying The Statement

A basic property of determinants is that if a square matrix has a row or column of zeros, the determinant is zero. Elementary row operations (which includes replacing a row by zeros) do not change the determinant. Hence, the given statement is true.

Key concepts

Square Matrix

A square matrix is a fundamental concept in linear algebra characterized by its equal number of rows and columns. The significance of square matrices, particularly in the context of determinants, stems from the fact that only for square matrices does the concept of a determinant apply. Determinants, which can be thought of as a scalar value, provide important information about the matrix such as whether its rows or columns are linearly independent, and if the matrix is invertible.

The determinant of a matrix is a single number that can be computed from the elements of a square matrix and offers insights into the properties of the matrix. It has applications in areas such as finding the area or volume of geometrical shapes and solving systems of linear equations. To put it simply, the determinant can be seen as a measure of how much the matrix will 'stretch' or 'shrink' space when used as a transformation.

Zero Row Property

The zero row property is a pivotal characteristic in the calculation of determinants. According to this property, if a square matrix has an entire row (or column) consisting of zeros, its determinant is immediately known to be zero. This property simplifies determinant calculations significantly in certain cases.

When dealing with a matrix featuring a zero row, the property saves computational effort since it eliminates the need for further exploration into its determinant – it will undoubtedly be zero. From a geometrical perspective, a zero row in a transformation matrix means that the transformation collapses the dimensionality of the space it operates on, hence no volume or area can be associated with it, leading to a determinant value of zero. This characteristic is essential for quick assessments of a matrix's invertibility, since a determinant of zero implies that the matrix cannot have an inverse.

Elementary Row Operations

Moving towards a more procedural aspect, elementary row operations are tools used to manipulate the rows of a matrix to achieve a desired form, often utilized in methods like Gaussian elimination. These operations are classified into three types: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another.

Despite their ability to transform a matrix, elementary row operations have a unique relationship with determinants: they impact the determinant of a square matrix in predictable ways. For example, swapping two rows reverses the sign of the determinant, and multiplying a row by a scalar multiplies the determinant by the same scalar. Notably, adding a multiple of one row to another row, which can create a zero row, does not affect the determinant's value. This means that while the elementary row operations are powerful tools for simplifying matrices, they must be applied with an understanding of how they will alter the determinant, if at all.