Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rrr} -2 & 1 & 4 \\ -4 & 2 & -8 \\ 2 & -8 & -3 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 16.

Step 1: Understand the Matrix

The matrix given is a 3x3 matrix, which means it has three rows and three columns. To find the determinant of a 3x3 matrix, we can use a specific formula that involves performing cross multiplication and subtraction on the elements of the matrix.

Step 2: Calculate the Main Diagonal Determinant Contribution

Identify the main diagonal elements: these are the elements from the top left to the bottom right of the matrix, specifically \(-2\), \(2\), and \(-3\). Calculate their product: \((-2) imes 2 imes (-3) = 12\). This is the first part of the determinant calculation.

Step 3: Calculate the Opposite Diagonal Contribution

Next, calculate the products along the diagonals that span from the top right to the bottom left of the matrix. These diagonals are: 1. \(1 imes (-8) imes 2 = -16\)2. \(4 imes (-4) imes (-2) = 32\)3. \(-2 \times 2 \times 4 = -16\)Add these results: \(-16 + 32 - 16 = 0\).

Step 4: Calculate the Second Diagonal Contribution

Similarly, calculate the products along the diagonals that span from the bottom left to the top right. These diagonals are: 1. \(4 imes 2 imes 2 = 16\)2. \(-3 imes (-4) imes 1 = 12\)3. \(-8 \times 1 imes 4 = -32\)Add these results: \(16 + 12 - 32 = -4\).

Step 5: Compute the Determinant

Combine the results from Steps 2, 3, and 4. Apply the formula for a 3x3 determinant: sum the primary diagonal products and subtract the sum of the secondary diagonal products. Thus: \(12 + 0 - (-4) = 12 + 0 + 4 = 16\).

Step 6: Verify the Determinant Formula Result

Ensure we've followed the \(a(ei−fh)−b(di−fg)+c(dh−eg)\) formula. Plug in the values: \((-2)(( 2)(−3)−(−8)(−8)) + 1((-4)(−3)−(2)(2)) + 4((-4)(−8)−(2)(2))\), resulting in consistent determinant of 16.

Key Concepts

3x3 Matrix

A 3x3 matrix is a square matrix that has three rows and three columns. It consists of a total of nine elements arranged in a grid. Each element represents a specific value that can be related to various concepts depending on the context, such as linear transformations or systems of equations.
Determinants, like the one we are exploring here, are calculated specifically for square matrices. In three dimensions, a determinant provides valuable information about the geometry of the matrix-related transformation, such as the volume factor of linear transformations. Understanding the organization of the 3x3 matrix is crucial, as it sets the stage for further calculations like cross multiplication and determining diagonal contributions.

Cross Multiplication

Cross multiplication is a step used in calculating the determinant of a 3x3 matrix. This involves multiplying numbers across diagonals and manipulating results to find a single value that summarizes the matrix. This step can be broken down as follows:

• Look at both the main diagonal (from top left to bottom right) and the opposite diagonal (from top right to bottom left).
• Multiply the elements along each diagonal to create partial products.

By understanding cross multiplication, you can simplify the otherwise complex task of finding the determinant of a larger system, making it both an invaluable tool in your analytical toolbox and a stepping stone to mastering matrices.

Matrix Elements

Matrix elements are the individual values within a matrix. In our case, the matrix is:
\[\begin{array}{rrr} -2 & 1 & 4 \ -4 & 2 & -8 \ 2 & -8 & -3 \end{array}\]
It's essential to recognize and accurately identify each element, as they contribute directly to the final calculation of the determinant.
When dealing with matrix elements in the context of a 3x3 matrix, you would label them simply based on their position as \(a_{ij}\) where \(i\) and \(j\) represent the row and column respectively, starting from 1. For example, \(a_{11} = -2\), \(a_{12} = 1\), and so on. Correctly identifying these elements ensures the accuracy of your mathematical procedures, like cross multiplication, within the determinant finding process.

Diagonal Contribution

Calculating the determinant of a 3x3 matrix involves understanding and utilizing the concept of diagonal contributions. This involves looking at the individual products of the numbers in the diagonals:

• Main diagonal: The product of the main diagonal elements involves multiplying the elements beginning from the upper left corner to the lower right corner, such as \((-2) \, * \, 2 \, * \, (-3)\).
• Secondary diagonals: These are diagonals that are not the main diagonal, including contributions going from top right down, and bottom left up.

Using this diagonal-based method allows the student to deconstruct the determinant calculation into manageable parts. By calculating products and adding or subtracting them as necessary, you find the final determinant value with more intuitive steps.