Problem 49 - Precalculus Mathematics for Calculus

Question

Evaluate the determinants. $$\left|\begin{array}{lllll} a & 0 & 0 & 0 & 0 \\ 0 & b & 0 & 0 & 0 \\ 0 & 0 & c & 0 & 0 \\ 0 & 0 & 0 & d & 0 \\ 0 & 0 & 0 & 0 & e \end{array}\right|$$

Step-by-Step Solution

Answer

Determinant = \(a \cdot b \cdot c \cdot d \cdot e\).

Step 1: Understanding the problem

We need to evaluate the determinant of a 5x5 matrix that is a diagonal matrix. A diagonal matrix has all its elements zero except those on the diagonal from the top left to the bottom right.

Step 2: Recognizing properties of diagonal matrices

For a diagonal matrix, the determinant is the product of the diagonal elements. This simplifies the calculation significantly, as we only need to focus on these elements.

Step 3: Identify diagonal elements

The diagonal elements of the given matrix are \(a, b, c, d,\) and \(e\). These values are found at positions (1,1), (2,2), (3,3), (4,4), and (5,5) respectively in the matrix.

Step 4: Calculating the determinant

Multiply the diagonal elements together to find the determinant: \(a \times b \times c \times d \times e\). This multiplication yields the determinant of the given matrix.

Key Concepts

Diagonal Matrix

A diagonal matrix is a special kind of matrix that is very important in mathematics. It has all its elements as zero except those situated diagonally from the top left corner to the bottom right corner. The diagonal elements are the only non-zero entries.
This configuration makes diagonal matrices simple but powerful.
They are easier to work with compared to other types of matrices. Diagonal matrices have many useful properties:

• Simplification: They simplify matrix computations like finding the determinant or inverse.
• Independence: Each diagonal entry acts independently without affecting others.
• Spectral Simplicity: They clearly show eigenvalues directly as the diagonal values.

Because of these traits, diagonal matrices are leveraged in simplifying complex mathematical operations and are foundational in linear algebra.

Matrix Properties

Matrices come with various properties that aid in simplifying operations.
Understanding these properties can often give shortcuts, especially with determinant calculations. Here's a list of some crucial matrix properties:

• Identity Property: An identity matrix, marked by ones on the diagonal and zeros elsewhere, leaves matrices unchanged when multiplied.
• Zero Matrix: A matrix filled entirely with zeros should have a determinant of zero.
• Triangular Matrix: If a matrix is upper or lower triangular (meaning all elements above or below the main diagonal are zero), its determinant is again the product of the diagonal elements.
• Transposition: The determinant remains unchanged when a matrix is transposed.

Recognizing these properties enables computations that are otherwise time-consuming.
Especially with diagonal and triangular matrices, these traits make calculations almost second nature.

5x5 Matrix Determinant Calculation

Evaluating the determinant of a 5x5 matrix can initially seem complex due to its size. However, when you're dealing with a diagonal matrix, the task becomes straightforward.When given a diagonal matrix like:\[ \begin{array}{ccccc} a & 0 & 0 & 0 & 0 \ 0 & b & 0 & 0 & 0 \ 0 & 0 & c & 0 & 0 \ 0 & 0 & 0 & d & 0 \ 0 & 0 & 0 & 0 & e \end{array} \]The calculation of its determinant is easier compared to non-diagonal matrices. Since the only non-zero elements are on its diagonal, the determinant is simply the product of these diagonal elements: \( a \times b \times c \times d \times e \)This method bypasses laborious methods like cofactor expansion. Understanding this property of diagonal matrices allows you to solve larger matrix determinant problemls efficiently by focusing only on the essential elements.