Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

A property of determinants is given \((A \text { and } B\) are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtaincd from \(A\) by interchanging two rows of \(A\) or by interchanging two columns of \(A,\) then \(|B|=-|A|\) $$\text { (a) }\left|\begin{array}{rrr}1 & 3 & 4 \\\\-7 & 2 & -5 \\\6 & 1 & 2\end{array}\right|=-\left|\begin{array}{rrr}1 & 4 & 3 \\\\-7 & -5 & 2 \\\6 & 2 & 1\end{array}\right|$$ $$\text { (b) }\left|\begin{array}{rrr}1 & 3 & 4 \\\\-2 & 2 & 0 \\\1 & 6 & 2\end{array}\right|=-\left|\begin{array}{rrr}1 & 6 & 2 \\\\-2 & 2 & 0 \\\1 & 3 & 4\end{array}\right|$$

Step-by-Step Solution

Answer

For both (a) and (b), the determinant of \(B\) is indeed the negation of the determinant of \(A\). Thus, the property of determinants \(|B|= - |A|\), when interchanging two rows or two columns in a square matrix \(A\) to obtain a new matrix \(B\), is verified.

Step 1: Identify the interchanged rows or columns

For (a) the columns are interchanged in matrix \(A\) to get the matrix \(B\). Specifically, column 2 and column 3 have been swapped. Similarly, for (b), row 2 and row 3 have been swapped in matrix \(A\) to get the matrix \(B\).

Step 2: Compute the determinant of \(A\)

To compute determinant, use the formula: \(\left| A \right| = a(ei−fh)−b(di−fg)+c(dh−eg)\) where \(A = \left[ {\begin{smallmatrix} a & b & c \\ d & e & f \\ g & h & i \end{smallmatrix}} \right]\). For (a), calculating gives \(|A| = 1*(2*2 - (-5)*1) - 3*(-7*2 - (-5)*6) + 4*(-7*1 - 6*2) = -40\). Similarly, for (b), \(|A| = 1*(2*2 - 0*6) -3*(-2*2 - 0*1) + 4*(-2*6 - 1*2) = 20\).

Step 3: Compute the determinant of \(B\)

Using same formula to compute the determinant of \(B\). For (a), \(|B| = -1*(2*2 - (-5)*1) + 4*(-7*2 - (-5)*6) - 3*(-7*1 - 6*2) = 40\). For (b), \(|B| = -1*(2*2 - 0*6) +2*(-2*2 - 0*1) - 2*(-2*6 - 1*2) = -20\)

Step 4: Verify the results

From the calculations, we can see that for both (a) and (b), \(|B|\) equals to the negation of \(|A|\). Thus, this verifies the property given in the exercise.

Key Concepts

Matrix Operations

Matrix operations form the backbone of various computational methods used in mathematics, physics, engineering, and data analysis. Fundamental operations include addition, subtraction, multiplication, and finding the determinant, among others. In the context of the original exercise, the determinant is the critical operation. When performing matrix operations, it is important to follow certain rules to ensure accurate results. For instance, only matrices of the same size can be added or subtracted, and the product of two matrices is only defined when the number of columns in the first matrix is equal to the number of rows in the second matrix. Matrix multiplication is not commutative, meaning that the order in which you multiply matrices matters.

Determinant of a Matrix

The determinant of a matrix is a scalar value that provides important insights into the properties of the matrix. It is denoted as \(|A|\) for a matrix \(A\) and can be calculated through various methods including, but not limited to, the method of cofactors, the Laplace expansion, and special formulas for 2x2 and 3x3 matrices.

The determinant can tell us if a matrix is invertible, with non-zero determinants indicating invertible matrices. Furthermore, the value of the determinant has geometric interpretations in the context of linear transformations represented by the matrix, such as the scale factor by which the transformation scales areas or volumes.

Graphing Utility Verification

In contemporary mathematics education, graphing utilities such as calculators and computer software are used to verify results obtained through manual calculations. This approach allows students and professionals to quickly confirm the accuracy of their solutions and understand how algebraic expressions correspond to geometric representations.

When verifying the results of matrix operations or determinants, a graphing utility can consume the matrix input and output the determinant, among other features. This technological verification is crucial for checking the results of exercises such as the ones provided, particularly when complex operations or large matrices are involved.

Interchanging Matrix Rows and Columns

Interchanging rows or columns within a matrix is one of the elementary row and column operations that can affect the determinant. In the provided exercise, it is stated that if a matrix \(B\) is derived from a matrix \(A\) by interchanging any two rows or columns, the determinant of \(B\) will be the negative of the determinant of \(A\). This is a fundamental property of determinants.

Understanding and utilizing properties such as these is essential when solving for determinants and working with linear systems. It simplifies larger computations and helps in finding solutions more efficiently, especially in theoretical proofs or when applying linear algebra to solve practical problems.