Problem 46 - Algebra and Trigonometry

Question

\(45-46=\) 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: Identify the Type of Matrix

Notice that the matrix is an upper triangular matrix because all the elements below the main diagonal are zeros.

Step 2: Determine the Determinant of Upper Triangular Matrix

For an upper triangular matrix, the determinant is calculated by multiplying all the diagonal elements together. Here, our diagonal consists of elements: \(a, a, a, a, a\).

Step 3: Solve the Product of Diagonal Elements

Calculate the product of the diagonal elements: \(a \times a \times a \times a \times a = a^5\).

Step 4: Write down the Determinant

Thus, the determinant of the matrix is \(a^5\).

Key Concepts

Understanding Upper Triangular Matrices

An upper triangular matrix is a special kind of square matrix. It's defined by having all non-zero elements located on or above the main diagonal, while all the elements below the diagonal are zeros. This specific structure gives it some interesting mathematical properties.

• The main diagonal is the line of elements stretching from the top-left to the bottom-right of the matrix, such as in an identity matrix.
• Due to its form, computations, especially those involving determinants, become more straightforward.
• In an upper triangular matrix, you can easily identify its structure by looking for the zeros below its main diagonal.

By recognizing these matrices, tasks like computing determinants are simplified. This makes them very handy in linear algebra.

The Role of Diagonal Elements

Diagonal elements play a crucial role in the computations involving matrices, particularly when it comes to calculating determinants. In a matrix, the diagonal elements are those that lie on the main diagonal, which stretches from the top-left corner to the bottom-right corner.

• For an upper triangular matrix, these diagonal elements are of extra importance because their product directly gives the determinant of the matrix.
• In our given matrix, the diagonal elements are all 'a', making them easy to identify and utilize.
• The ease of computation with these elements is a major advantage, as demonstrated in the step-by-step solution of our problem.

These elements essentially control the matrix's determinant, simplifying what can otherwise be a complex computation.

Matrix Multiplication Simplified

Matrix multiplication involves combining two matrices to produce a third one. It is central to many mathematical operations in linear algebra. While our problem focuses on determinants, understanding multiplication can clarify how matrices interact. When focusing on a square matrix, or especially upper triangular matrices, the multiplication process becomes a foundation for other operations:

• For any two matrices, the inner dimensions must match (e.g., a 2x3 matrix can multiply a 3x4 matrix).
• The elements from the row of the first matrix are multiplied by the corresponding elements from the column of the second matrix.
• These products are then summed to produce a single element in the resulting matrix.

This might seem different from simply multiplying diagonal elements for the determinant, but understanding the multiplication process helps visualize how matrices interact internally.