Problem 29 - Precalculus Mathematics for Calculus

Question

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$\left|\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 0 & 2 & 4 & 6 & 8 \\ 0 & 0 & 3 & 6 & 9 \\ 0 & 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & 0 & 5 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is 120.

Step 1: Identify the Matrix Type

The given matrix is an upper triangular matrix, meaning all the elements below the main diagonal are zero. Recall that the determinant of an upper triangular matrix is the product of its diagonal elements.

Step 2: Identify Diagonal Elements

The diagonal elements of the matrix are 1, 2, 3, 4, and 5.

Step 3: Calculate Determinant

Multiply the diagonal elements to find the determinant: \[ 1 \times 2 \times 3 \times 4 \times 5 = 120 \]

Step 4: Conclusion

Since the matrix is upper triangular and we've multiplied all the diagonal elements, the determinant of the matrix is 120.

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a special kind of matrix where all the elements below the main diagonal are zero. In simpler terms, if you look at a square matrix, draw an imaginary line cutting across it from the top left corner to the bottom right corner, each element below this line should be zero for the matrix to be upper triangular. The given matrix is a perfect example, where all non-diagonal elements in the lower section are zero.
Understanding this concept is crucial because it simplifies determinant evaluation significantly.

• It reduces computation because the process involves simple multiplication of the diagonal elements.
• Recognizing an upper triangular matrix can quickly lead you to the solution, saving time.

Hence, whenever you encounter a matrix where elements below the diagonal are zero, remember you can use this short-cut to find the determinant.

Diagonal Elements

Diagonal elements in a matrix are those positioned from the top left to the bottom right in a square matrix. In other words, for a given matrix \( a_{ij} \), the diagonal elements have the same row and column indices, \( i = j \). These elements play a significant role in determinant calculation when dealing with upper triangular matrices.
In our case, these diagonal elements are:

• 1
• 2
• 3
• 4
• 5

The beauty of diagonal elements in an upper triangular matrix is that they simplify the determinant calculation. Instead of dealing with a complex series of operations, you multiply these diagonal numbers together to find the determinant.

Matrix Operations

Matrix operations encompass various mathematical methods employed to manipulate matrices, such as addition, subtraction, and multiplication. However, when evaluating the determinant of a matrix, we often focus on transforming matrices into simpler forms, like upper triangular matrices, to make the calculation easier.
In this context, row and column operations can simplify matrices or transform them into an upper triangular form. Yet here, our matrix was already in its simplest form. The only operation required was the multiplication of diagonal elements. Understanding these operations is key in precalculus mathematics, as they solve numerous problems more efficiently.
Learning when to apply these operations can significantly reduce the complexity of mathematical problems involving matrices and determinants.

Precalculus Mathematics

Precalculus mathematics lays the foundation for more advanced mathematical concepts, including calculus. It often introduces students to matrices, their properties, and operations, which are crucial for understanding systems of equations and transformations in various dimensions.
The topic of evaluating determinants of matrices, particularly understanding the properties of special matrices like upper triangular matrices, is part of this foundational knowledge. Recognizing patterns and applying simple operations, like multiplying diagonal elements, helps in building this foundational skill. Mastery in precalculus can greatly enhance your efficiency and success in calculus and other mathematical disciplines.
Thus, understanding these preliminary topics well will give you a strong base to handle complex problems in upper-level math courses with ease.