Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rrr} 2 & -1.6 & 3.1 \\ 1.1 & 3 & -8 \\ -9.3 & 0 & 2 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -17.03.

Step 1: Identify the Matrix Order

The given matrix is a 3x3 matrix. The determinant of a 3x3 matrix can be calculated using a specific formula involving the elements of the matrix.

Step 2: Use the Determinant Formula for 3x3 Matrix

For a given 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is calculated as: \( \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).

Step 3: Substitute the Values from the Matrix

Using the formula, substitute the values from the matrix: \( a = 2 \), \( b = -1.6 \), \( c = 3.1 \), \( d = 1.1 \), \( e = 3 \), \( f = -8 \), \( g = -9.3 \), \( h = 0 \), and \( i = 2 \).

Step 4: Calculate the Adjoint Components

Calculate each of the minor products: 1. \( ei - fh = 3 \times 2 - (-8) \times 0 = 6 \)2. \( di - fg = 1.1 \times 2 - (-8) \times (-9.3) = 2.2 - 74.4 = -72.2 \)3. \( dh - eg = 1.1 \times 0 - 3 \times (-9.3) = 27.9 \).

Step 5: Apply the Determinant Formula

Substitute the calculated components into the formula: \( \det(A) = 2 \times 6 - (-1.6) \times (-72.2) + 3.1 \times 27.9 \). This simplifies to: \( \det(A) = 12 - 115.52 + 86.49 \).

Step 6: Compute the Final Determinant Value

Simplify the expression: \( 12 - 115.52 + 86.49 = -17.03 \). Thus, the determinant of the matrix is \(-17.03\).

Key Concepts

3x3 Matrix

A 3x3 matrix is a square array consisting of three rows and three columns, with each position in the matrix filled by a number. These matrices are used widely in mathematics, physics, engineering, and computer science to solve linear equations, perform transformations, and represent data systematically.

For example, the matrix given in the exercise consists of three rows

• Row 1: [2, -1.6, 3.1]
• Row 2: [1.1, 3, -8]
• Row 3: [-9.3, 0, 2]

Each cell in the matrix is referred to by its position, like the first number in the first row is 2. Recognizing this structure is essential since operations like determinants are calculated based on how numbers are arranged in these rows and columns.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrices and their operations. It involves processes such as addition, multiplication, and finding determinants or inverses of matrices.

When working with a matrix, understanding basic operations is crucial:

• **Addition/subtraction:** Combine matrices by adding or subtracting each corresponding element.
• **Multiplication:** More complex; involves taking rows from the first matrix and columns from the second, performing element-wise multiplication and summing the results.
• **Determinants:** Unique to square matrices, they provide scalar values that offer insights into the properties of the matrix.

These operations help solve systems of linear equations, find eigenvectors, and perform various transformations. Being comfortable with algebraic manipulation of matrices opens up numerous practical applications.

Determinant Calculation

Determinants are mathematical tools that help understand properties like invertibility and the volume scaling factor when a linear transformation represented by the matrix is applied.

For a 3x3 matrix, as shown in the example, the determinant is calculated using a specific formula. It interacts with all elements of the matrix:

• **Diagonals and Products:** The determinant formula uses specific products from diagonals of the matrix, emphasizing how each subcomponent contributes.
• **Adjoint Components:** Calculating these involves multiplying and summing elements. For instance, a diagonal product would be calculated as in step 4 of the exercise.
• **Calculating the Determinant:** Combine these products with signs alternating to ensure an accurate determinant, simplifying complex attributes of the matrix into a single number that reflects its scale and certain transformation properties.

Understanding determinants aids in working with linear transformations, evaluating systems of linear equations, and determining matrix inversions. They summarize critical aspects of the matrix into tangible, understood values.