Problem 50 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The determinant is \(a^5\).

Step 1: Understand the Matrix Structure

The matrix is a lower triangular matrix with all diagonal elements equal to \(a\) and all elements above the diagonal are zero.

Step 2: Apply Triangular Matrix Rule for Determinants

In a square matrix, if all the elements above or below the main diagonal (excluding diagonal) are zero, the matrix is said to be triangular. The determinant of a triangular matrix is the product of its diagonal entries.

Step 3: Calculate the Determinant

Since each diagonal entry of the matrix is \(a\) and the matrix is 5x5, the determinant is the product of 5 \(a\)'s: \(a \cdot a \cdot a \cdot a \cdot a = a^5\).

Key Concepts

Triangular Matrix

A triangular matrix is a special type of square matrix where all the elements are either above or below the main diagonal are zero. This special structure can either be upper triangular or lower triangular.

• An **upper triangular matrix** has all elements above the diagonal as non-zero, while those below are zero.
• A **lower triangular matrix** has all elements below the diagonal as non-zero, while those above are zero.

Triangular matrices are crucial in various mathematical applications due to their straightforward properties, particularly when calculating determinants. They simplify the process, making it easier to compute than with general matrices.

Lower Triangular Matrix

A lower triangular matrix is one where all the elements above the main diagonal are zero. This means that if you imagine walking down the staircase of numbers from the top left to the bottom right, you only encounter numbers on stairs or below.

• Each row begins with several zeroes until the diagonal element.
• All entries above any given element are zero.

In the context of the exercise, the matrix given is a lower triangular matrix because all elements above the diagonal are zero. This makes calculating its determinant much simpler.

Matrix Determinant Properties

Matrix determinant is a value that can tell us a lot about a matrix, including whether it is invertible, and more. Determinants have several important properties that make calculations easier. One such property deals with triangular matrices.For any triangular matrix:

• The determinant is simply the product of its diagonal elements. This property streamlines calculations since you don't need to perform complex row operations.
• This means for a 5x5 lower triangular matrix with diagonal elements all equal to \(a\), the determinant will be \(a^5\).

Understanding this property is key, as it greatly reduces computation time and effort when dealing with triangular matrices.