Problem 61 - 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

The determinant is the product \( a \times b \times c \times d \times e \).

Step 1: Identify the Matrix Type

The given matrix is a 5x5 diagonal matrix. A diagonal matrix has non-zero elements only on its main diagonal, with all off-diagonal elements being zero.

Step 2: Formula for Determinant of a Diagonal Matrix

The determinant of a diagonal matrix is the product of its diagonal elements. This simplifies the process significantly since we don't have to perform any row or column operations.

Step 3: Multiply Diagonal Elements

To find the determinant, multiply each of the diagonal elements together: \( a \times b \times c \times d \times e \).

Step 4: Write the Determinant Value

Substitute the product of the diagonal elements: \( a \times b \times c \times d \times e \) into the determinant formula. This product represents the final solution of the determinant for this matrix.

Key Concepts

Diagonal Matrix

Understanding what a diagonal matrix is can significantly simplify the process of finding a determinant. A diagonal matrix is a special type of matrix where:
- All the elements outside the main diagonal are zeros.
- The main diagonal (from the top left to the bottom right) can have non-zero values.

In essence, the only numbers we need to focus on for operations are the ones on this diagonal line. This unique structure makes certain matrix operations, like finding determinants, much easier and faster! For instance, imagining a 5x5 matrix like the one shown, you'd only consider `a, b, c, d,` and `e` located down its diagonal for many calculations, while all other positions contain zeros.

Matrix Determinant Formula

The determinant itself is a special value that can tell us quite a bit about the matrix. For instance, it can reveal whether a system of linear equations has a unique solution. Typically, calculating a determinant for larger matrices can be complex and involves different methods, like row reduction or expansion by minors. However, diagonal matrices offer a straightforward path.

For a diagonal matrix, we simply multiply the diagonal values together to find the determinant. This is known as the "matrix determinant formula" for diagonal matrices. So, for a diagonal matrix:
\[\text{Determinant} = a \times b \times c \times d \times e\]
This formula is powerful because it bypasses many convoluted steps from traditional matrix determinant calculations. Imagine using this simple multiplication instead of an arduous row operation or cofactor expansion.

Product of Diagonal Elements

The "product of diagonal elements" concept is specific to diagonal matrices. It simplifies determinant calculation vastly. Rather than engaging in complicated matrix operations, you just perform a simple multiplication of the diagonal terms. This is why knowing whether a matrix is diagonal or not initially saves a lot of effort.

Here's how it works in practice: if we have a diagonal matrix with diagonal elements labeled as `a, b, c, d,` and `e`, to find the determinant, we compute:
\[\text{Determinant} = a \times b \times c \times d \times e\]
This straightforward process is a fantastic demonstration of how particular matrix structures can make mathematical tasks less of a chore. The resulting product, or the determinant, provides a concise numerical summary of the matrix's essential properties. Thus, understanding the role of diagonal elements in determinant calculations not only streamlines the process but also enhances comprehension of matrix theory at large.