Problem 27 - Algebra for College Students

Question

Evaluate each \(3 \times 3\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrr}2 & 3 & -4 \\ 4 & 6 & -1 \\ -6 & 1 & -2\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is -140.

Step 1: Understand the Determinant

The determinant of a 3x3 matrix can be found using the formula: \( \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \) where \( A = \left|\begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array}\right| \). In this matrix, \( a = 2, b = 3, c = -4, d = 4, e = 6, f = -1, g = -6, h = 1, i = -2 \).

Step 2: Calculate Each Minor Expression

Replace the variables \( (ei - fh) \), \( (di - fg) \), and \( (dh - eg) \) using the given values: - \( ei - fh = 6(-2) - (-1)(1) = -12 + 1 = -11 \). - \( di - fg = 4(-2) - (-1)(-6) = -8 - 6 = -14 \). - \( dh - eg = 4(1) - 6(-6) = 4 + 36 = 40 \).

Step 3: Substitute the Minors into Determinant Formula

Using the formula \( \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \), substitute the values: - For \( a(ei − fh) = 2(-11) = -22 \). - For \( -b(di − fg) = -3(-14) = 42 \). - For \( c(dh − eg) = -4(40) = -160 \).

Step 4: Calculate Final Determinant Value

Add the calculated values from Step 3: \( -22 + 42 - 160 = -140 \). Hence, the determinant of the matrix is \( -140 \).

Key Concepts

Matrix Determinant Calculation

Calculating the determinant of a 3x3 matrix involves a specific process that helps obtain a unique numerical value. This value provides important properties about the matrix related to its invertibility and how it transforms space.
In the original exercise, the problem involves determining the determinant using the formula for a 3x3 matrix:

• The formula is: \( \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \)
• This involves selecting a row or column (commonly the first row) and applying the three terms of the formula.
• Each term corresponds to a minor of the matrix, which is determined by eliminating the corresponding row and column of each element in the chosen row or column.

By calculating each of the minor expressions accurately, you can follow through with the substitution into the formula to find the determinant, as demonstrated in the solution steps. This procedure, once mastered, is essential in more complex linear algebra computations.

Properties of Determinants

Determinants possess various properties that are useful in simplifying calculations and understanding matrix behavior.
Some of the key properties that can be leveraged while working with determinants include:

• Row and Column Swap: Swapping two rows or two columns of a matrix changes the sign of the determinant.
• Row/Column Multiplication: Multiplying all elements of a row or column by a scalar multiplies the determinant by the same scalar.
• Row/Column Addition: Adding a multiple of one row to another does not change the determinant.
• Triangular Matrices: The determinant is the product of the diagonal elements when the matrix is triangular.

Recognizing these properties simplifies the computation and provides deeper insights into the interplay of matrix elements.

Linear Algebra Concepts

In the realm of linear algebra, solving determinants like in this exercise serves important purposes.
The determinant of a matrix gives insights into the geometrical and algebraic characteristics of the transformation described by the matrix.

• Invertibility: If a determinant is zero, the matrix is singular and does not have an inverse.
• Area and Volume Transformation: In two or three dimensions, the absolute value of the determinant provides the scaling factor of areas or volumes when applying the matrix transformation.
• Linear Independence: A zero determinant suggests that the vectors forming the rows or columns of the matrix are linearly dependent.

Understanding these concepts enhances the appreciation of the determinant beyond mere computation.