Problem 16 - Advanced Engineering Mathematics

Question

In Problems 15-18, evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{B}=\left(\begin{array}{rrr} 0 & 0 & a_{13} \\ 0 & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is 0.

Step 1: Recognize the Upper Triangular Matrix

In the given matrix \( \mathbf{B} \), identify that it resembles an upper triangular matrix, where all elements below the main diagonal are zeros. The main diagonal in this matrix is \([0, a_{22}, a_{33}]\).

Step 2: Check the Matrix's Structure

Confirm that the given matrix structure maintains properties of an upper triangular matrix by verifying that all elements below the diagonal are zero. Our matrix \( \mathbf{B} \) has this exact property, with all elements below the diagonal being zero, making it an upper triangular matrix.

Step 3: Calculate the Determinant

For an upper triangular matrix, the determinant is the product of the diagonal elements. Therefore, the determinant of \( \mathbf{B} \) is calculated as:\[\text{det}(\mathbf{B}) = 0 \times a_{22} \times a_{33} = 0\]

Step 4: Verify the Calculation

Verify that the determinant calculation is consistent with properties of determinants. Since one of the diagonal elements is zero (in our case, the first element), the product of these elements is zero, which makes the determinant zero.

Key Concepts

Understanding Upper Triangular Matrices

An upper triangular matrix is a special type of square matrix where all the elements below the main diagonal are zero. This structure is significant when calculating determinants because it provides a simplified approach. For example, in matrix \( \mathbf{B} \), every element below the main diagonal—running from the top left to the bottom right—is zero.
This makes it easy to spot, as the zero elements create a distinct pattern, forming a triangle of non-zero elements above the diagonal. This pattern is what gives the matrix its name. Recognizing an upper triangular matrix helps us compute determinants quickly and efficiently without using complex operations like cofactor expansion.

Matrix Structure and Its Importance

The structure of a matrix is crucial for understanding its properties, especially when it comes to solving problems like determinant calculation. The matrix \( \mathbf{B} \) we've been analyzing is a 3x3 matrix, which is a simple, small square matrix perfect for demonstrating certain mathematical concepts.

• In an upper triangular matrix, you check for zeros below the main diagonal. This is a unique feature that drastically simplifies many calculations.
• In mathematical operations, recognizing this structure prevents unnecessary complexity, such as when using extensive matrix reduction techniques.

Always start by examining the pattern of zeros; this practice saves time and provides insights into the behavior of the matrix under various operations.

Determinant Properties with Practical Examples

Determinants provide vital information about a matrix and have several interesting properties, particularly when considering upper triangular matrices. One of the most useful properties is that the determinant of an upper triangular matrix is simply the product of its diagonal elements.
In the case of our example matrix \( \mathbf{B} \), the diagonal elements are \([0, a_{22}, a_{33}]\). When we multiply these diagonal elements, the presence of a zero in the diagonal means the entire product, and hence the determinant, equals zero.

• This property holds for all upper triangular matrices, vastly simplifying the determinant calculation process.
• If any of the elements on the main diagonal are zero, the determinant will always be zero. This highlights the importance of diagonal elements in determining the overall matrix property.

Understanding these properties allows for quick evaluations of matrix behavior, aiding in both theoretical and applied mathematics.