Problem 27 - Precalculus Mathematics for Calculus

Question

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$\left|\begin{array}{llll} 0 & 0 & 4 & 6 \\ 2 & 1 & 1 & 3 \\ 2 & 1 & 2 & 3 \\ 3 & 0 & 1 & 7 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is -18.

Step 1: Identify Zero-Containing Rows or Columns

Examine the given matrix to see if there is a row or column with zeros that would simplify computation. The first row has two zeros.

Step 2: Expand Along the First Row

Given that the first row has two zeros, expand the determinant along the first row. This simplifies calculations because terms involving these columns will be zero. The matrix becomes:\[(-1)^{1+3} \cdot 4 \cdot \left| \begin{array}{ccc}2 & 1 & 3 \2 & 1 & 3 \3 & 0 & 7\end{array} \right|+ (-1)^{1+4} \cdot 6 \cdot \left| \begin{array}{ccc}2 & 1 & 1 \2 & 1 & 2 \3 & 0 & 1\end{array} \right|\]

Step 3: Evaluate the First 3x3 Minor

Compute the determinant of the first 3x3 minor:\[\left| \begin{array}{ccc}2 & 1 & 3 \2 & 1 & 3 \3 & 0 & 7\end{array} \right|\]Using row operations, subtract the first row from the second row:\[\left| \begin{array}{ccc}2 & 1 & 3 \0 & 0 & 0 \3 & 0 & 7\end{array} \right|\]Notice the second row is all zeros, hence the determinant is zero.

Step 4: Evaluate the Second 3x3 Minor

Compute the determinant of the second 3x3 minor:\[\left| \begin{array}{ccc}2 & 1 & 1 \2 & 1 & 2 \3 & 0 & 1\end{array} \right|\]Use row operations to simplify. Subtract the first row from the second:\[\left| \begin{array}{ccc}2 & 1 & 1 \0 & 0 & 1 \3 & 0 & 1\end{array} \right|\]Now apply the formula for a 3x3 determinant:\[= 2(0 \cdot 1 - 1 \cdot 0) - 1(0 \cdot 1 - 1 \cdot 3) + 1(0 \cdot 0 - 1 \cdot 3) = 3\]

Step 5: Combine Results

Combine the results from the first expansion using the computed minors:\[D = 4 \cdot 0 - 6 \cdot 3 = -18\]

Step 6: Conclusion: Determinant Value

The determinant of the matrix is \(-18\).

Key Concepts

Understanding Row Operations

Row operations are essential when calculating the determinant of a matrix. They simplify matrices without changing the determinant except when a row is multiplied by a scalar, in which case the determinant is also multiplied by that scalar. There are three types of row operations:

• Row swapping: This involves switching the positions of two rows. It changes the sign of the determinant but not the absolute value.
• Row multiplication: Multiplying a row by a nonzero constant, which multiplies the determinant by the same constant.
• Row addition: Adding or subtracting a multiple of one row from another. This operation does not change the determinant.

When evaluating the determinant of a matrix, row operations can be used to create zeros in a row or column, simplifying further calculations.
In step 3 of the solution, row subtraction is used to make the second row all zeros, making it easy to see that the determinant of that minor is zero. Understanding when and how to apply these operations is key to mastering determinant evaluation.

Column Operations and Their Use

Column operations are similar to row operations, and while they are less often highlighted, they offer valuable techniques to simplify determinant evaluation. Just like row operations, these include:

• Column swapping: Interchanging two columns, which changes the sign of the determinant.
• Column multiplication: Multiplying all elements of a column by a constant, multiplying the determinant by that constant.
• Column addition: Adding a multiple of one column to another without affecting the determinant value.

Although the provided solution for the original exercise primarily uses row operations, column operations are equally valid and can similarly lead to simplifications in determinant calculations. They are especially useful when working with matrices with many zeros in a column, guiding you toward more efficient computation.

Breaking Down a 3x3 Matrix

Understanding 3x3 matrices is a fundamental step in learning linear algebra. These matrices are often used because they are complex enough to demonstrate broader mathematical principles but small enough to grasp manually. A 3x3 matrix takes the following form:\[\begin{bmatrix}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33}\end{bmatrix}\]The formula for calculating the determinant of such a matrix is:\[det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]This calculation involves breaking the matrix into smaller "minors." In the provided solution, after creating a 3x3 matrix by removing the row and column containing a zero, row operations helped further simplify the process. Additionally, using zeros effectively through understanding minor rules and Laplace expansion is vital. The simplicity and elegance of the 3x3 determinant formula make it a go-to for practicing determinant evaluation. It's a gateway to handling larger matrices successfully.