Problem 5 - Intermediate Algebra

Question

For Problems \(1-10\), use expansion by minors to evaluate each determinant. (Objective 1) $$ \left|\begin{array}{rrr} -3 & -2 & 1 \\ 5 & 0 & 6 \\ 2 & 1 & -4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is \(-41\).

Step 1: Select a Row for Expansion

For expansion by minors, you can select any row or column. It's often easiest to choose a row or column with the most zeros, but here, no zeros are present. We'll expand along the first row: \\(-3, -2, \) and \(1\).

Step 2: Calculate the Minor for the First Element

The first element in the first row is \(-3\). To find its minor, remove the row and column containing \(-3\). The remaining determinant is: \\(\left|\begin{array}{rr} 0 & 6 \ 1 & -4 \end{array}\right|\).

Step 3: Evaluate the Determinant of the Minor

Calculate the 2x2 determinant: \\(0(-4) - 6(1) = 0 - 6 = -6\).

Step 4: Calculate the Cofactor for the First Element

The cofactor of the \(-3\) is calculated using the formula \( (-1)^{i+j} \times \text{minor}\). For element \(-3\), \(i=1\) and \(j=1\), hence: \\((-1)^{2} imes (-6) = 1 \times -6 = -6\).

Step 5: Calculate the Minor for the Second Element

Move on to the next element \(-2\) in the first row. Remove its row and column to find the new 2x2 determinant: \\(\left|\begin{array}{rr} 5 & 6 \ 2 & -4 \end{array}\right|\).

Step 6: Evaluate the Determinant of the Minor

Compute this 2x2 determinant: \\(5(-4) - 6(2) = -20 - 12 = -32\).

Step 7: Calculate the Cofactor for the Second Element

Calculate the cofactor of \(-2\) using \((-1)^{i+j} \times \text{minor}\). Here, \(i=1\), \(j=2\), giving us: \\((-1)^{3} \times (-32) = -1 \times -32 = 32\).

Step 8: Calculate the Minor for the Third Element

For the last element \(1\) in the first row, remove its row and column: \\(\left|\begin{array}{rr} 5 & 0 \ 2 & 1 \end{array}\right|\).

Step 9: Evaluate the Determinant of the Minor

Work out the 2x2 determinant: \\(5(1) - 0(2) = 5 - 0 = 5\).

Step 10: Calculate the Cofactor for the Third Element

The cofactor for \(1\) is found with \((-1)^{i+j} \times \text{minor}\), where \(i=1\) and \(j=3\): \\((-1)^{4} \times 5 = 1 \times 5 = 5\).

Step 11: Calculate the Determinant Using the Cofactors

Now, substitute back the cofactors into the determinant: \\(-3(-6) + (-2)(32) + 1(5) = 18 - 64 + 5\).

Step 12: Final Calculation and Result

Sum up the calculations: \\(18 - 64 + 5 = -41\). \Thus, the determinant of the matrix is \(-41\).

Key Concepts

Determinant Evaluation

Evaluating the determinant of a matrix can seem complex at first, but by using the expansion by minors method, it's much simpler. This method involves selecting a row or column from the matrix and breaking down the complex determinant into smaller, more manageable parts.
Begin by choosing which row or column to expand on. A row or column with zeroes is typically preferred, as it simplifies calculations. However, if none are available, like in our exercise, you can choose any row or column. In this case, the first row \(-3, -2, 1\) makes a convenient choice.
For each element in the row, you'll find the corresponding minor and cofactor. By repeating this process for each element, the full determinant becomes a sum of these simpler calculations. The simplicity and power of this method lie in systematically reducing complexity, making determinant evaluation much more approachable.

Matrix Algebra

Matrix algebra is a critical component of linear algebra, and understanding how to work with matrices is essential for this problem. A matrix is essentially a rectangular array of numbers, and its determinant provides valuable information about the matrix itself, such as whether it has an inverse.
To use matrices effectively:

• Understand the basic operations such as addition, subtraction, and multiplication.
• Learn the properties of matrices such as their dimensions and types (e.g., square matrices).
• Practice finding determinants and inverses, which are common tasks in matrix algebra.

By mastering these simple operations and concepts, you can tackle more advanced topics, such as solving systems of linear equations or analyzing eigenvalues, all of which require a good grasp of matrix algebra concepts.

Cofactor Calculation

The calculation of cofactors is a fundamental step in the process of determinant evaluation. Cofactors are used to adjust the minors by incorporating a sign change, depending on the element's position in the matrix.
Here's how to calculate a cofactor:

• Identify the row and column of the element in question, with indices given by \(i\) and \(j\).
• Compute the minor by removing the corresponding row and column from the matrix.
• Apply the sign factor \((-1)^{i+j}\) to the minor. This crucial step ensures the correct sign, based on the element's position.

The cofactor not only assists in evaluating the determinant but is also key in calculating the adjugate matrix, which subsequently helps in finding the inverse of a matrix. Understanding cofactors is thus an important skill in linear algebra, enabling you to deeply engage with the structure and properties of matrices.