Problem 75 - Precalculus: Mathematics for Calculus

Question

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

Determinant is 0 due to zero rows/columns, repeated rows/columns, or linearly dependent rows/columns.

Step 1: Analyze a Matrix with a Zero Row/Column

If a matrix has a row or column consisting entirely of zeros, the determinant of the matrix is zero. This is because when expanding the determinant using cofactor expansion along the zero row or column, each term in the expansion involves a multiplication by zero, resulting in a total determinant of zero.

Step 2: Evaluate a Matrix with Repeated Rows/Columns

When a matrix has two identical rows or two identical columns, the determinant is zero. This is derived from properties of the determinant, where swapping two identical rows or columns does not change the determinant but conceptually, this would negate it. Thus, the determinant must be zero since it contradicts the invariance under these swaps.

Step 3: Consider a Matrix with Proportional Rows/Columns

If one row of a matrix is a multiple of another, or one column is a multiple of another, the determinant of the matrix is zero. This arises because creating linear dependency by having proportional rows or columns means the matrix's rank is less than full, leading to a determinant of zero as a consequence of dependency.

Key Concepts

Zero Row or Column

Imagine you are given a matrix where one entire row or one entire column is filled only with zeros. What can we infer about this matrix? It turns out that the determinant of such a matrix is always zero. This happens due to the nature of the mathematical operation called cofactor expansion. When you expand the determinant along a row or column, each element is multiplied by a subdeterminant. However, if every element of the row or column is zero, every term in this expansion becomes zero, because zero multiplied by anything is still zero.
This makes the whole determinant equal zero. Thus, the presence of a zero row or column makes calculating the determinant straightforward—it's always zero.

Repeated Rows or Columns

Now, let's consider what happens when you encounter a matrix with two identical rows or two identical columns. These are known as repeated rows or columns. Such matrix configurations also result in a determinant of zero. Here's why: When you swap two rows or two columns in a regular matrix, the determinant changes sign.
However, if the rows or columns are identical and we swap them, the matrix looks the same, suggesting a paradox. Swapping shouldn't change the determinant sign, leading us to a logical contradiction. Thus, for the determinant to remain coherent with these properties, it must be zero when rows or columns are repeated. This ensures consistency with the mathematical properties of determinants.

Proportional Rows or Columns

Finally, consider matrices with rows or columns that are purely proportional. This means one row or column is a scaled version of another. For instance, one row might be twice as big as another. When this occurs, the matrix is said to be linearly dependent.
Proportional rows or columns imply there's redundancy in the information presented by the matrix. For any row that is a multiple of another, the linear system it represents isn't fully independent. As a consequence, such matrices do not have a full rank. In linear algebra, when a matrix lacks full rank, its determinant is zero. This dependency suggests a collapse in the multi-dimensional "space" the matrix explores, leading to a zero determinant.