Problem 46 - Precalculus: Mathematical 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: Analyze the Matrix

We have a 5x5 upper triangular matrix, where all diagonal elements are equal to 'a' and the rest above the diagonal are also 'a'. A unique feature in an upper triangular matrix is that its determinant is the product of its diagonal elements.

Step 2: Apply the Determinant Rule for Triangular Matrices

For an upper triangular matrix, the determinant is the product of the diagonal elements. Here, those diagonal elements are: \(a, a, a, a, a\). Therefore, the determinant is \(a imes a imes a imes a imes a\).

Step 3: Simplify the Expression

Calculate the product of the diagonal elements. Each is equal to 'a', and the product is \(a^5\). Thus, the determinant of the given matrix is \(a^5\).

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix where all the elements below the main diagonal are zero. In simpler terms, if you look at the matrix, you will see zeros filling the entire lower half beneath the diagonal. This pattern creates an easily recognized shape that looks like a triangle pointing upwards.

Upper triangular matrices are commonly used in solving linear algebra problems because they have properties that simplify calculations, like finding determinants. For any size of an upper triangular matrix, whether it is 2x2, 5x5, or even larger, only the elements on or above the main diagonal are non-zero. This unique feature makes such matrices easier to handle in computations, especially when it comes to matrix operations such as finding their inverse or solving equations.

• Recognize the pattern of zeros below the diagonal.
• Understand that calculations are simplified by focusing solely on the diagonal elements.
• Use this type of matrix to streamline solving complex linear equations.

Diagonal Elements

Diagonal elements are the entries of a matrix that run from the top left to the bottom right corner. In a 5x5 matrix, this would involve five specific entries, one from each row and column. For an upper triangular matrix, these diagonal elements play a crucial role.

They are central to calculations involving determinants since their product determines the determinant value in a triangular matrix context. Knowing the value of each diagonal element allows us to perform easy determinant calculations without needing to consider non-diagonal entries. In our exercise, since all diagonal elements have the same value 'a', it simplifies finding the matrix's determinant to multiplying 'a' by itself as many times as there are diagonal elements.

• Identify diagonal elements quickly in any matrix.
• Realize their significance in determinant calculations.
• Save time and effort by focusing on these elements.

Determinant Properties

The determinant of a matrix is a special number that can provide insights into various properties of the matrix, such as invertibility and solution uniqueness of linear equations. For triangular matrices, such as the upper triangular matrix, a fascinating property emerges: the determinant is simply the product of its diagonal elements.

This property is not only time-saving but also simplifies calculations, especially in larger matrices. Unlike general matrix determinant calculations, which can involve complex operations, the rule for triangular matrices only requires a straightforward multiplication of diagonal values.

In the exercise example, because the matrix is upper triangular, we found its determinant by multiplying each diagonal element, all equal to 'a', resulting in the product being as simple as evaluating this sequence: \[a \times a \times a \times a \times a = a^5\]

• Lean on determinant properties for quick calculations.
• Understand how matrix shape affects determinant evaluation.
• Apply knowledge of determinant properties in linear algebra problems.