Problem 35 - Algebra for College Students

Question

Use the appropriate property of determinants from this section to justify each true statement. Do not evaluate the determinants. \(\left|\begin{array}{rrr}4 & 7 & 9 \\ 6 & -8 & 2 \\ 4 & 3 & -1\end{array}\right|=-\left|\begin{array}{rrr}4 & 9 & 7 \\ 6 & 2 & -8 \\ 4 & -1 & 3\end{array}\right|\)

Step-by-Step Solution

Answer

Swapping two columns of a matrix changes the sign of its determinant.

Step 1: Identify the Property

Recognize that the two matrices differ by columns being permuted. The first matrix has its columns differently ordered compared to the second matrix.

Step 2: Apply the Determinant Property

Apply the property that swapping two columns (or rows) of a matrix multiplies the determinant by -1. Here, swapping two columns results in the negative sign before the determinant.

Step 3: Verify the Column Swap

Note that in the second matrix, the first column (4, 6, 4) remains unchanged, while the second and third columns, (7, -8, 3) and (9, 2, -1), have swapped positions compared to the first matrix.

Key Concepts

Determinant Properties

Determinants offer key insights into a matrix's characteristics in linear algebra. They provide a single value that can help determine whether a matrix is invertible. A matrix is invertible if and only if its determinant is non-zero. Determinants tell us about the volume scaling factor of a linear transformation represented by a matrix. When discussing determinant properties, it is crucial to recognize several rules:

• If two rows or columns of a matrix are identical, its determinant is zero.
• Swapping two rows or columns changes the sign of the determinant (multiplies it by -1).
• A matrix with a row or column full of zeroes has a determinant of zero.
• Multiplying a row or column by a scalar multiplies the determinant by that scalar.

Understanding these properties helps in quickly assessing matrices and performing algebraic manipulations without evaluating the determinants directly. This simplifies solving large and complex problems, especially when verifying matrix transformations.

Matrix Column Swap

When you perform a matrix column swap, you are rearranging the columns of the matrix. In terms of determinants, swapping two columns (or rows) results in a sign change of the determinant of the matrix. This is a fundamental property and is used to simplify calculations or solve problems without full computation.

Consider a 3x3 matrix with columns labeled as C1, C2, and C3. If you swap C2 and C3, the new determinant will be the negative of the original. This change occurs because the determinant can be thought of as a sum of products, and swapping C2 and C3 results in products being reorganized in a different order, impacting the overall sign.

• Original: \( ext{det}(C1, C2, C3) \)
• After swap: \( ext{det}(C1, C3, C2) = - ext{det}(C1, C2, C3) \)

Understanding this swap property is crucial in linear algebra operations, where ordered arrangements can influence matrix evaluation outcomes.

Linear Algebra

Linear algebra is a branch of mathematics concerned with vector spaces and linear mappings between them. It is foundational for various disciplines, including computer science, engineering, and physics. At its core, linear algebra involves solving systems of linear equations, performing operations on matrices, and exploring vector spaces. Matrices are central to linear algebra, serving as compact representations of systems.

Key concepts in linear algebra include:

• Vector Spaces: Collections of vectors that can be added together and multiplied by scalars.
• Matrices: Rectangular arrays of numbers or functions representing linear transformations or systems of linear equations.
• Determinants: Scalar values that describe certain properties of matrices, such as invertibility and orientation.
• Eigenvalues and Eigenvectors: Scalars and vectors that show directions that remain unchanged under linear transformations.

Linear algebra allows us to model and solve physical and abstract problems, engage in multi-dimensional analysis, and explore transformations in space.