Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}-1 & 2 & 4 \\\4 & -8 & -16 \\\3 & 0 & 5\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 0.

Step 1: Identify Matrix

Let's identify the matrix given in the problem: \[ A = \begin{bmatrix} -1 & 2 & 4 \ 4 & -8 & -16 \ 3 & 0 & 5 \end{bmatrix} \] We are tasked with finding the determinant of this 3x3 matrix using determinant theorems.

Step 2: Determine Matrix Size

The matrix \( A \) is a 3x3 matrix because it has 3 rows and 3 columns.

Step 3: Use Determinant Formula for 3x3 Matrix

To calculate the determinant of a 3x3 matrix \[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \], use the formula: \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Step 4: Substitute Matrix Values into Formula

Substitute the values from matrix \( A \) into the formula: \[ a = -1, b = 2, c = 4, d = 4, e = -8, f = -16, g = 3, h = 0, i = 5 \] Substitute these into the formula: \[ \det(A) = -1((-8)(5) - (-16)(0)) - 2((4)(5) - (-16)(3)) + 4((4)(0) - (-8)(3)) \]

Step 5: Calculate Each Term Separately

Calculate each expression inside the parentheses:- \( -1((-8)(5) - (-16)(0)) = -1(-40) = 40 \)- \( -2((4)(5) - (-16)(3)) = -2(20 + 48) = -2(68) = -136 \)- \( 4((4)(0) - (-8)(3)) = 4(0 + 24) = 4(24) = 96 \)

Step 6: Add the Results

Sum up the calculated terms to find the determinant: \[ \det(A) = 40 - 136 + 96 \]

Step 7: Calculate the Final Determinant

Perform the calculation: \[ 40 - 136 + 96 = 0 \]. Thus, the determinant of matrix \( A \) is 0.

Key Concepts

3x3 Matrix

A 3x3 matrix is a mathematical structure that consists of three rows and three columns, forming a grid with nine elements in total. Matrices, in general, are used to handle complex mathematical operations involving transformations, system of equations, and more.
A 3x3 matrix, specifically, looks like this:

• The first row might consist of the elements \( a, b, c \)
• The second row contains \( d, e, f \)
• The third row has \( g, h, i \)

This configuration is represented as follows:\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]The elements \( a, b, c, \ldots, i \) are numbers that can be real or complex depending on the context of the problem. Having a 3x3 matrix means you're working within a scope that allows for two-dimensional transformations and is essential for many applications in science and engineering.

Determinant Theorems

Determinants are numerical values calculated from a square matrix like a 3x3 matrix, which can be used for numerous purposes such as solving systems of linear equations, determining matrix invertibility, and more.
Several determinant theorems can simplify the calculation or understanding of determinants:

• Row and Column Operations: Swapping two rows or columns changes the determinant's sign; multiplying a row by a scalar multiplies the determinant by the same scalar.
• Determinants of Triangular Matrices: If a matrix is upper (or lower) triangular, the determinant is the product of its diagonal elements.
• Addition of Rows: Adding a multiple of one row to another doesn't change the determinant.

For a 3x3 matrix, a primary method to find the determinant is expanding across a row or a column, using the cofactor expansion method, which involves minor matrices and their corresponding cofactors.

Matrix Calculation

Calculating the determinant of a 3x3 matrix involves a specific formula derived from expansion by minors. The formula for a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]The steps to calculate the determinant are as follows:

• Identify Elements: Extract the elements in positions \( a, b, c, d, e, f, g, h, i \) from the matrix.
• Substitute Values: Insert these elements into the formula.
• Simplify Expressions: Perform the arithmetic operations within the parentheses, and carefully multiply terms.
• Combine Results: Add the calculated results to get the determinant.

In our example, once each substitution and calculation is complete, the resulting determinant was zero, indicating the matrix A is singular and does not have an inverse.