Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 2 & 4 & 6 \\ 0 & 3 & 1 \\ 0 & 0 & -5 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is -30.

Step 1: Identify the Matrix Type

The given matrix is an upper triangular matrix, meaning all entries below the main diagonal are zero. For such matrices, the determinant is simply the product of the diagonal entries.

Step 2: Calculate the Determinant

Calculate the determinant of this matrix by multiplying the diagonal entries: \( 2 \times 3 \times -5 = -30 \)

Step 3: Confirm Results with a Graphing Utility

At this stage, use a graphing calculator or other graphing utility you have available to calculate the determinant and confirm your result. In most graphical utilities, you just need to input the matrix and use the determinant function.

Key Concepts

Upper Triangular Matrix

Understanding an upper triangular matrix is crucial when studying linear algebra. An upper triangular matrix is a special kind of square matrix where all the elements below the main diagonal are zero. Visually, it resembles a triangle filled with non-zero entries at the top, while the bottom part is clear.

Consider this matrix as a building with floors, where you can only move upwards from the ground floor, never downwards. That's how data travels in an upper triangular matrix— only across and up, never down.

In simple terms, for a matrix \( A \) of size \( n \times n \), it is considered upper triangular if \( A_{ij} = 0 \) for all \( i > j \). In such matrices, finding the determinant doesn't require complicated procedures. You multiply the elements on the diagonal—the 'elevator' of the building, stretching from the bottom left to the top right. So, if your matrix is \( \left[\begin{array}{ccc} a & b & c \ 0 & d & e \ 0 & 0 & f \end{array}\right] \), the determinant would simply be \( a \times d \times f \) since those are the 'floors' of the building.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is like a treasure hunt for the determinant of any square matrix—piece by piece, you work your way towards the prize. It's especially useful when dealing with matrices that aren't as straightforward as upper triangular ones.

This method involves picking a row or column—typically the one with the most zeros because it makes calculations easier—and breaking down the determinant into smaller, manageable parts. Here's how it works: You look at each element \( a_{ij} \) in your chosen row or column, and for each element, you calculate its cofactor. The cofactor is \( (-1)^{i+j} \) times the determinant of the submatrix that you get when you remove the \( i \)th row and \( j \)th column from the original matrix.

Then, you sum up the products of the original elements and their respective cofactors to find the determinant of the matrix. It sounds like a lot of steps, but with practice, cofactor expansion becomes a powerful tool in your math toolkit. It is especially helpful when a matrix isn't neatly organized into an upper triangular format.

Graphing Utility

In modern mathematics, a graphing utility is like the trusted sidekick to your mathematical heroics—there to confirm your answers and give you visual insights into complex problems. It isn't just for graphing equations but also comes armed with advanced features, including calculating determinants.

Using a graphing utility, you can input any square matrix and instruct the tool to calculate its determinant for you. It's a real time-saver, particularly for large matrices or when you want to double-check your manual calculations. To use this feature, you usually select the matrix input option, fill in the matrix, and hit the function that prompts the utility to give you the determinant.

For example, when working with the matrix from our exercise, you would enter the values into the utility, select the determinant operation, and let technology do the rest. This step is great for verification and ensures that your manual calculations are on the right track. Plus, it demonstrates the practicality of math in real-world applications—showcasing that beyond pen and paper, there are tools that mathematicians, scientists, and engineers use every day.