Problem 36 - A Graphical Approach to College Algebra

Question

Evaluate each determinant. $$\operatorname{det}\left[\begin{array}{rrrr}4 & 5 & -1 & -1 \\\2 & -3 & 1 & 0 \\\\-5 & 1 & 3 & 9 \\\0 & -2 & 1 & 5\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -134.

Step 1: Identify the Matrix

The determinant we need to evaluate is a 4x4 matrix:\[\begin{bmatrix}4 & 5 & -1 & -1 \ 2 & -3 & 1 & 0 \ -5 & 1 & 3 & 9 \ 0 & -2 & 1 & 5\end{bmatrix}\]

Step 2: Choose a Row or Column for Expansion

To simplify the determinant calculation, choose the last row for cofactor expansion since it contains zeroes which make calculations easier.

Step 3: Expand the Determinant Along the Chosen Row

Use the formula for determinant expansion along row 4:\[det(A) = 0 imes C_{41} + (-2) imes C_{42} + 1 imes C_{43} + 5 imes C_{44}\]Since the element in the first position is zero, it contributes nothing to the determinant.

Step 4: Calculate Cofactor \(C_{42}\)

The minor for \(C_{42}\) (ignoring row 4 and column 2) is:\[\begin{vmatrix} 4 & -1 & -1 \ 2 & 1 & 0 \ -5 & 3 & 9 \end{vmatrix}\]Apply cofactor expansion along the first row.

Step 5: Expand \(C_{42}\) Minor

Calculate the determinant using the first row for cofactor expansion:\[4 \cdot \begin{vmatrix} 1 & 0 \ 3 & 9 \end{vmatrix} - (-1) \cdot \begin{vmatrix} 2 & 0 \ -5 & 9 \end{vmatrix} - (-1) \cdot \begin{vmatrix} 2 & 1 \ -5 & 3 \end{vmatrix}\]Evaluate each 2x2 determinant.

Step 6: Evaluate 2x2 Determinants for \(C_{42}\)

1. \(\begin{vmatrix} 1 & 0 \ 3 & 9 \end{vmatrix} = 1(9) - 3(0) = 9\)2. \(\begin{vmatrix} 2 & 0 \ -5 & 9 \end{vmatrix} = 2(9) - 0(-5) = 18\)3. \(\begin{vmatrix} 2 & 1 \ -5 & 3 \end{vmatrix} = 2(3) - 1(-5) = 11\) Substitute back to get: \[4 \cdot 9 + 1 \cdot 18 + 1 \cdot 11 = 36 + 18 + 11 = 65\]

Step 7: Calculate Cofactor \(C_{43}\)

Using the matrix without row 3 and column 3, find:\[\begin{vmatrix} 4 & 5 & -1 \ 2 & -3 & 0 \ 0 & -2 & 5 \end{vmatrix}\]Apply cofactor expansion along the first row.

Step 8: Expand \(C_{43}\) Minor

Calculate the determinant using the first row:\[4 \cdot \begin{vmatrix} -3 & 0 \ -2 & 5 \end{vmatrix} - 5 \cdot \begin{vmatrix} 2 & 0 \ 0 & 5 \end{vmatrix} - (-1) \cdot \begin{vmatrix} 2 & -3 \ 0 & -2 \end{vmatrix}\]Evaluate each 2x2 determinant.

Step 9: Evaluate 2x2 Determinants for \(C_{43}\)

1. \(\begin{vmatrix} -3 & 0 \ -2 & 5 \end{vmatrix} = (-3)(5) - 0(-2) = -15\)2. \(\begin{vmatrix} 2 & 0 \ 0 & 5 \end{vmatrix} = 2(5) = 10\)3. \(\begin{vmatrix} 2 & -3 \ 0 & -2 \end{vmatrix} = 2(-2) - (-3)(0) = -4\)Substitute back to get:\[4 \cdot (-15) + 5 \cdot 10 + 1 \cdot -4 = -60 + 50 - 4 = -14\]

Step 10: Calculate Cofactor \(C_{44}\)

Using the matrix without row 3 and column 4, find:\[\begin{vmatrix} 4 & 5 & -1 \ 2 & -3 & 1 \ -5 & 1 & 3 \end{vmatrix}\]Apply cofactor expansion along the first row.

Step 11: Expand \(C_{44}\) Minor

Calculate the determinant using the first row:\[4 \cdot \begin{vmatrix} -3 & 1 \ 1 & 3 \end{vmatrix} - 5 \cdot \begin{vmatrix} 2 & 1 \ -5 & 3 \end{vmatrix} - (-1) \cdot \begin{vmatrix} 2 & -3 \ -5 & 1 \end{vmatrix}\]Evaluate each 2x2 determinant.

Step 12: Evaluate 2x2 Determinants for \(C_{44}\)

1. \(\begin{vmatrix} -3 & 1 \ 1 & 3 \end{vmatrix} = (-3)(3) - 1(1) = -10\)2. \(\begin{vmatrix} 2 & 1 \ -5 & 3 \end{vmatrix} = 2(3) - 1(-5) = 11\)3. \(\begin{vmatrix} 2 & -3 \ -5 & 1 \end{vmatrix} = 2(1) - (-3)(-5) = 2 - 15 = -13\)Substitute back to get:\[4 \cdot (-10) + 5 \cdot 11 + 1 \cdot -13 = -40 + 55 - 13 = 2\]

Step 13: Combine the Cofactors

Using the calculated cofactors, evaluate the determinant:\[-2 \times 65 + 1 \times (-14) + 5 \times 2 = -130 - 14 + 10 = -134\]

Step 14: Output the Determinant Value

The determinant of the matrix is \(-134\). Thus, \[\operatorname{det}\begin{bmatrix} 4 & 5 & -1 & -1 \ 2 & -3 & 1 & 0 \ -5 & 1 & 3 & 9 \ 0 & -2 & 1 & 5 \end{bmatrix} = -134 \]

Key Concepts

4x4 matrix

A 4x4 matrix is simply a square arrangement of numbers or elements with 4 rows and 4 columns. Each element is defined by its row and column position and is essential for operations such as determinant calculation. Knowing how to interpret and work with such matrices is a foundation in linear algebra.

When faced with a 4x4 matrix, the task of finding its determinant may seem daunting due to its size. However, the process becomes manageable if broken down into smaller tasks, applying known algebraic techniques like cofactor expansion. This large matrix is systematically reduced into smaller, more straightforward pieces to provide a single scalar value, the determinant. This value is crucial as it tells us about certain properties of the matrix, such as invertibility.

cofactor expansion

Cofactor expansion, also known as Laplace's expansion, is a method used to calculate the determinant of a matrix. While directly evaluating a larger matrix determinant like 4x4 can be complex, cofactor expansion simplifies this by breaking the matrix into smaller components.

To apply a cofactor expansion, choose one row or one column. It's beneficial to choose a row or column with zeroes, as it lessens the computation needed. In this process, each element is multiplied by its cofactor, which is calculated from the determinant of a smaller matrix formed by excluding the row and column of the current element.

• Select a row or column.
• Determine each element's cofactor.
• Multiply each element by its cofactor and sum up the results.

Cofactor expansion is an iterative technique, which allows us to handle matrices of any size by reducing them to simpler 2x2 or 3x3 forms.

2x2 matrix determinant

The 2x2 matrix is the simplest kind of square matrix where evaluating the determinant is very straightforward. It is, therefore, a key building block for calculating determinants of larger matrices, such as using cofactor expansion.

Given a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant can easily be calculated using the formula:
\[\operatorname{det}\begin{bmatrix} a & b \ c & d \end{bmatrix} = ad - bc\]This quick calculation provides the scalar value which is used as a part of cofactor expansion. The 2x2 determinant reflects the matrix's basic properties, similar to larger matrices, such as whether it's invertible or not. Understanding 2x2 determinants is crucial for advancing in matrix operations and algebraic problem-solving.

algebra problem solving

Algebra problem-solving involves various strategies and methods to understand and manipulate mathematical expressions and equations to find solutions. Matrices and their determinants form a significant part of algebra, especially in linear algebra, where solving these problems is necessary for understanding more intricate systems of equations.

These problems require a step-by-step approach, breaking down the exercise into manageable parts like identifying the matrix structure, employing determinant properties, and implementing cofactor expansion. Each step helps in simplifying complex algebraic processes. As one progresses with these methods, algebra problem solving becomes an essential tool in both academic studies and applied mathematics, highlighting the importance of logical reasoning and mathematical techniques.