Problem 34 - Intermediate Algebra

Question

Show that \(\left|\begin{array}{rrr}1 & -1 & 2 \\ 2 & 3 & -1 \\ -1 & 2 & 4\end{array}\right|=-\left|\begin{array}{rrr}-1 & 1 & 2 \\ 3 & 2 & -1 \\ 2 & -1 & 4\end{array}\right|\). Make a conjecture about the result of interchanging two columns of a determinant.

Step-by-Step Solution

Answer

Interchanging two columns negates the determinant's value.

Step 1: Calculate the first determinant

First, let's compute the determinant of the matrix \( A = \begin{pmatrix} 1 & -1 & 2 \ 2 & 3 & -1 \ -1 & 2 & 4 \end{pmatrix}\). Use the formula for a \(3 \times 3\) determinant: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]where - \(a = 1, b = -1, c = 2, d = 2, e = 3, f = -1, g = -1, h = 2, i = 4 \).Substituting these values, we have:\[ \det(A) = 1(3 \cdot 4 - (-1) \cdot 2) - (-1)(2 \cdot 4 - (-1) \cdot -1) + 2(2 \cdot 2 - 3 \cdot (-1)) \]Now, compute each term:\[ 3 \cdot 4 - (-1) \cdot 2 = 12 + 2 = 14 \]\[ 2 \cdot 4 - (-1)(-1) = 8 - 1 = 7 \]\[ 2 \cdot 2 - 3 \cdot (-1) = 4 + 3 = 7 \]So, the determinant becomes:\[ \det(A) = 1(14) - (-1)(7) + 2(7) = 14 + 7 + 14 = 35 \].

Step 2: Calculate the second determinant

Now, we need to compute the determinant of the matrix \( B = \begin{pmatrix} -1 & 1 & 2 \ 3 & 2 & -1 \ 2 & -1 & 4 \end{pmatrix} \). Apply the same formula:\[ \det(B) = a(ei - fh) - b(di - fg) + c(dh - eg) \]where - \(a = -1, b = 1, c = 2, d = 3, e = 2, f = -1, g = 2, h = -1, i = 4 \).Substitute these values:\[ \det(B) = -1(2 \cdot 4 - (-1) \cdot (-1)) - 1(3 \cdot 4 - (-1) \cdot 2) + 2(3 \cdot -1 - 2 \cdot 2) \]Compute each term:\[ 2 \cdot 4 - (-1)(-1) = 8 - 1 = 7 \]\[ 3 \cdot 4 - (-1) \cdot 2 = 12 + 2 = 14 \]\[ 3 \cdot (-1) - 2 \cdot 2 = -3 - 4 = -7 \]So, the determinant becomes:\[ \det(B) = -1(7) - 1(14) + 2(-7) = -7 - 14 - 14 = -35 \].

Step 3: Compare the determinants and make a conjecture

From the calculations, we have \( \det(A) = 35 \) and \( \det(B) = -35 \). It can be observed that by comparing these determinants, the absolute values are the same, but have opposite signs.Conjecture: Interchanging two columns of a determinant results in negating the value of the determinant.

Key Concepts

Properties of Determinants

Determinants possess unique mathematical properties that simplify computations and enhance understanding of matrices. The determinant, a scalar value, can describe certain characteristics of a matrix which remain consistent, even after transformations like row or column operations. One key property of determinants is linearity, which states that if any row or column of a matrix is a linear combination of other rows or columns, the determinant is zero.

• Determinants reflect the scaling factor by which a linear transformation changes area or volume. When a matrix’s determinant is zero, it indicates that the transformation collapses the space into a dimension lower than expected, usually forming a line or point.
• Another crucial property is permutation. Multiplying the rows or columns of a matrix by a scalar results in the determinant being multiplied by the same scalar.

Understanding these properties is fundamental for grasping larger concepts, such as matrix invertibility and solving linear equations using matrices.

Matrix Algebra

Matrix algebra is a branch of mathematics that focuses on the operations and properties of matrices. It involves the manipulation of matrices in various ways to solve linear equations, transform geometric entities, and model complex systems through mathematical representations.

• Key operations in matrix algebra include addition, scalar multiplication, and matrix multiplication, each defined uniquely compared to traditional arithmetic operations. These operations allow the creation and manipulation of systems of linear equations at scale.
• One crucial operation is finding the determinant, a process essential for understanding whether a square matrix is invertible. A non-zero determinant implies that the matrix has an inverse, valuable for computations involving matrix division and system solutions.

Matrix algebra provides tools for complex calculations, significantly impacting fields such as computer science, engineering, and physics by modeling and solving multi-variable problems.

Column Interchange Effect on Determinants

Interchanging two columns of a determinant leads to a specific outcome: the determinant's value is negated, meaning its sign changes while the absolute value remains constant. This property emerges directly from how determinants are computed, as the arrangement and orientation of matrix entries influence the calculation.

• When columns are swapped, each permutation changes the sign of the determinant due to the alternating signs used in the determinant's expansion formula. This effectively reflects a reversal of orientation within the mathematical space represented by the matrix.
• This concept is crucial when manipulating matrices, especially in strategies like Gaussian elimination, where transforming matrices without changing determinants excessively is valuable for finding polynomial roots and eigenvalues.

Grasping the impact of column interchange on determinants assists in navigating complex matrices more effectively, enhancing algebraic manipulation skills and understanding fundamental geometric interpretations of linear transformations.