Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rrr} 5 & 1 & -1 \\ 2 & 3 & 1 \\ 3 & -6 & -3 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 15.

Step 1: Understand the Determinant of a 3x3 Matrix

To find the determinant of a 3x3 matrix, we use the rule of Sarrus or the cofactor expansion. Here, we will use the cofactor expansion method along the first row. A 3x3 matrix is given by \(A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\). According to the method, the determinant \(|A|\) is calculated as \(a(ei - fh) - b(di - fg) + c(dh - eg)\).

Step 2: Identify Elements for Cofactor Expansion

From the given matrix \(\begin{bmatrix} 5 & 1 & -1 \ 2 & 3 & 1 \ 3 & -6 & -3 \end{bmatrix}\), identify the elements \(a = 5\), \(b = 1\), and \(c = -1\) from the first row. These elements will be used in the cofactor expansion formula.

Step 3: Calculate Each Minor

Calculate each of the minors needed for the cofactor expansion:- For the element \(5\), the minor is \(\left(3 \times -3\right) - \left(1 \times -6\right)\).- For the element \(1\), the minor is \(\left(2 \times -3\right) - \left(1 \times 3\right)\).- For the element \(-1\), the minor is \(\left(2 \times -6\right) - \left(3 \times 3\right)\).

Step 4: Evaluate Each Minor

- The minor for \(5\) is \((-9) - (-6) = -3\).- The minor for \(1\) is \((-6) - 3 = -9\).- The minor for \(-1\) is \((-12) - 9 = -21\).

Step 5: Apply the Cofactor Expansion Formula

Use the values from Steps 2 and 4 in the cofactor expansion formula. The determinant \(|A|\) is calculated as:\[|A| = 5(-3) - 1(-9) + (-1)(-21)\]Simplify each term:- \(5(-3) = -15\)- \(1(-9) = 9\)- \(-1(-21) = 21\)

Step 6: Sum the Evaluated Terms

Combine the results from Step 5:\[-15 + 9 + 21\]Calculate the sum:\[-15 + 9 = -6\]\[-6 + 21 = 15\]

Step 7: Write the Determinant

The determinant \(|A|\) of the matrix is 15. This is the result of evaluating the cofactor expansion and combining all terms.

Key Concepts

Cofactor Expansion

The cofactor expansion is an essential method to calculate the determinant of a 3x3 matrix. This technique involves expressing the determinant in terms of the matrix's elements and their corresponding cofactors. Cofactor expansion allows us to choose any row or column for the expansion, simplifying calculations.

In the case of our example, we opted to use the first row for cofactor expansion. The rule is to multiply each element by its respective cofactor and then sum these results. Cofactors are derived by taking the minor and multiplying it by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the element.

• The formula is: \(|A| = aC_{1} + bC_{2} + cC_{3}\), where \(C_n\) denotes each cofactor.
• For the first element 5, we calculated its minor and determined its cofactor, then followed this same process for the others.

This method relies heavily on understanding both cofactors and minors as core components of the process.

Minor of a Matrix

Minors lay the groundwork for calculating cofactors. A minor is a determinant of a 2x2 submatrix formed by eliminating one row and one column from the matrix. Each minor corresponds to a specific element of the original matrix.

Let's look at how it's done for our example:

• For the element 5 in the first row and first column, its minor involves removing the first row and the first column.
• The result is the submatrix: \[\begin{bmatrix} 3 & 1 \ -6 & -3 \end{bmatrix}\]
• We find its determinant as:\((3 \times -3) - (1 \times (-6)) = -9 + 6 = -3\)

This process is repeated for the other elements in the row. Minors transform the 3x3 matrix into more manageable 2x2 determinants, directly aiding in calculating cofactors.

Matrix Algebra

Matrix algebra encompasses a wide array of operations involving matrices, and it is vital in linear algebra. Understanding matrix algebra can simplify complex operations like solving systems, finding determinants, and working on transformations.

Determinants are one piece of this larger puzzle, instrumental in various applications:

• They help in solving linear systems by determining if solutions exist.
• They play a role in calculating the inverse of matrices, which requires a non-zero determinant.
• Used extensively in transforming geometrical interpretations.

In our exercise, matrix algebra's methods provide a structured way to tackle problems with matrices, leveraging operations like cofactor expansion for practical solutions. Engaging deeply with these methods enhances proficiency in handling matrices and solidifies foundational knowledge in mathematics.