Problem 23 - College Algebra

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 matrix elements

The given matrix is a 3x3 matrix:\[ A = \begin{bmatrix} 2 & -1.6 & 3.1 \ 1.1 & 3 & -8 \ -9.3 & 0 & 2 \end{bmatrix} \] Identify elements: \[ a_{11} = 2, \, a_{12} = -1.6, \, a_{13} = 3.1, \ a_{21} = 1.1, \, a_{22} = 3, \, a_{23} = -8, \ a_{31} = -9.3, \, a_{32} = 0, \, a_{33} = 2 \]

Step 2: Apply the 3x3 determinant formula

The determinant of a 3x3 matrix \( A \) is given by:\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \] Substituting the elements from the matrix into the formula gives:

Step 3: Calculate each cofactor

Compute the cofactors individually:- \( a_{11}(a_{22}a_{33} - a_{23}a_{32}) = 2(3 \times 2 - (-8) \times 0) = 2(6) = 12 \)- \( a_{12}(a_{21}a_{33} - a_{23}a_{31}) = -1.6(1.1 \times 2 - (-8) \times (-9.3)) = -1.6(2.2 - 74.4) = -1.6(-72.2) = 115.52 \)- \( a_{13}(a_{21}a_{32} - a_{22}a_{31}) = 3.1(1.1 \times 0 - 3 \times (-9.3)) = 3.1(0 + 27.9) = 86.49 \)

Step 4: Sum the results to find the determinant

Combine the calculated cofactors:\[ \text{det}(A) = 12 - 115.52 + 86.49 = -17.03 \] The determinant of the matrix is \(-17.03\).

Key Concepts

3x3 matrix

A 3x3 matrix is a special type of square matrix that contains three rows and three columns. This means it has exactly 9 elements in total. Each element in the matrix is usually represented by a specific notation such as \(a_{ij}\), where \(i\) indicates the row number and \(j\) indicates the column number.
For example, the matrix in our exercise is:

• Row 1: \(a_{11}=2, \, a_{12}=-1.6, \, a_{13}=3.1\)
• Row 2: \(a_{21}=1.1, \, a_{22}=3, \, a_{23}=-8\)
• Row 3: \(a_{31}=-9.3, \, a_{32}=0, \, a_{33}=2\)

3x3 matrices are common in many fields such as physics and engineering for representing systems of equations or transformations. Working with 3x3 matrices involves various operations, and one of the key features is calculating the determinant, which can inform us about the properties of the system represented by the matrix.

Cofactor expansion

Cofactor expansion, also known as Laplace expansion, is a method used to calculate the determinant of a matrix. For a 3x3 matrix, this involves selecting a particular row or column and calculating the determinant based on the elements and their cofactors.
In our exercise, we expand along the first row. The determinant of the matrix \( A \) is calculated as follows:

• Select row or column: The first row is chosen: \(a_{11}, \, a_{12}, \, a_{13}\).
• Compute cofactors:
  • \(C_{11} = a_{11}(a_{22}a_{33} - a_{23}a_{32})\).
  • \(C_{12} = -a_{12}(a_{21}a_{33} - a_{23}a_{31})\).
  • \(C_{13} = a_{13}(a_{21}a_{32} - a_{22}a_{31})\).
  These values are summed with alternating signs: \(det(A) = C_{11} - C_{12} + C_{13}\).

Cofactor expansion highlights the structure of the matrix and provides deeper insights into its composition by examining the smaller parts of the matrix, which are called minors. It also helps in understanding the relationship between different elements of the matrix.

Matrix operations

Matrix operations are operations that can be performed on matrices, such as addition, subtraction, multiplication, and computing determinants. In this exercise, we are primarily focused on the determinant operation. The determinant is a scalar value that can be computed from the elements of a square matrix.
Computing a determinant for a 3x3 matrix involves the following:

• Identify all elements: Record down each value in the matrix.
• Apply the determinant formula: Use the specific formula for 3x3 matrices to break down the determinant using cofactors.
• Perform arithmetic operations: Carry out step-by-step computations to reach the final value.

The determinant provides important information about a matrix, such as whether it is invertible. If the determinant is non-zero, the matrix is invertible, meaning it has an inverse matrix. Conversely, a zero determinant tells us that the matrix does not have an inverse, and therefore, it represents a system without a unique solution.