Problem 20 - Advanced Engineering Mathematics

Question

In Problems 19 and 20, verify that \(\operatorname{det} \mathbf{A}=\operatorname{det} \mathbf{A}^{T}\) for the given matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 4 \\ 1 & 0 & 5 \\ 7 & 2 & -1 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of \( \mathbf{A} \) is equal to the determinant of \( \mathbf{A}^T \), both are 136.

Step 1: Define the Determinant

The determinant of a matrix \( \mathbf{A} \) can be found using several methods. For a 3x3 matrix, we can use the formula for the determinant based on the elements of the first row: \[ \det(\mathbf{A}) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where \( \mathbf{A} = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \).

Step 2: Identify Elements of Matrix \( \mathbf{A} \)

For the matrix \( \mathbf{A} = \begin{pmatrix} 2 & 3 & 4 \ 1 & 0 & 5 \ 7 & 2 & -1 \end{pmatrix} \), the elements are: \( a=2, b=3, c=4, d=1, e=0, f=5, g=7, h=2, i=-1 \).

Step 3: Calculate Determinant of \( \mathbf{A} \)

Substitute the elements into the determinant formula: \[ \det(\mathbf{A}) = 2(0 \times -1 - 5 \times 2) - 3(1 \times -1 - 5 \times 7) + 4(1 \times 2 - 0 \times 7) \] which simplifies to: \[ \det(\mathbf{A}) = 2(0 + 10) - 3(-1 - 35) + 4(2) \].

Step 4: Solve the Expression

Evaluate the expression from Step 3: \[ \det(\mathbf{A}) = 2(10) + 3(36) + 8 = 20 + 108 + 8 = 136 \]. This gives \( \det(\mathbf{A}) = 136 \).

Step 5: Create the Transpose of \( \mathbf{A} \)

The transpose of a matrix \( \mathbf{A} \), denoted \( \mathbf{A}^T \), is achieved by swapping rows with columns. Given \( \mathbf{A} = \begin{pmatrix} 2 & 3 & 4 \ 1 & 0 & 5 \ 7 & 2 & -1 \end{pmatrix} \), its transpose is \( \mathbf{A}^T = \begin{pmatrix} 2 & 1 & 7 \ 3 & 0 & 2 \ 4 & 5 & -1 \end{pmatrix} \).

Step 6: Calculate Determinant of \( \mathbf{A}^T \)

Using the same method as before, find \( \det(\mathbf{A}^T) \) by substituting its elements: \[ \det(\mathbf{A}^T) = 2(0 \times -1 - 5 \times 2) - 1(3 \times -1 - 5 \times 4) + 7(3 \times 2 - 0 \times 4) \].

Step 7: Solve Determinant of \( \mathbf{A}^T \)

Evaluate the determinant of \( \mathbf{A}^T \): \[ \det(\mathbf{A}^T) = 2(0 + 10) - 1(-3 - 20) + 7(6) \]. Simplifying gives \[ \det(\mathbf{A}^T) = 20 + 23 + 42 = 136 \]. Therefore, \( \det(\mathbf{A}^T) = 136 \).

Step 8: Conclude the Verification

Since we have \( \det(\mathbf{A}) = 136 \) and \( \det(\mathbf{A}^T) = 136 \), it confirms that \( \det(\mathbf{A}) = \det(\mathbf{A}^T) \), verifying the property for the given matrix \( \mathbf{A} \).

Key Concepts

Transpose of a Matrix

When working with matrices, a fundamental concept is the transpose of a matrix. Transposing a matrix involves swapping its rows and columns, essentially flipping it over its diagonal. If you have a matrix \( \mathbf{A} = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), its transpose, denoted as \( \mathbf{A}^T \), would be \( \begin{pmatrix} a & d & g \ b & e & h \ c & f & i \end{pmatrix} \). Notice how the first row of \( \mathbf{A} \) becomes the first column of \( \mathbf{A}^T \), and so on for the rest of the matrix.

Transposing a matrix does not change the size of the matrix—it remains a 3x3 matrix if it starts as a 3x3. While transposing, ensure you carefully switch the correct elements to avoid mistakes. This operation proves useful in several mathematical applications, including solving systems of linear equations and finding determinants.

3x3 Matrix

A 3x3 matrix is a square matrix that contains three rows and three columns. It is often used in various mathematical and engineering applications because of its manageable size and complexity. The structure of a 3x3 matrix is such that it contains nine elements in total. These elements can be numbers, algebraic expressions, or other mathematical entities, depending on the context.

Each element of a 3x3 matrix is indexed by its row and column position. For example, in matrix \( \mathbf{A} \) given by \( \begin{pmatrix} 2 & 3 & 4 \ 1 & 0 & 5 \ 7 & 2 & -1 \end{pmatrix} \), the element "2" is in the first row, first column, and "-1" is in the third row, third column.

3x3 matrices are particularly important because they represent transformations in three-dimensional space, such as rotation, scaling, and translation. In linear algebra, they are crucial for solving determinant-based problems, eigenvalue calculations, and more.

Determinant Calculation

The determinant of a square matrix like a 3x3 matrix is a special number that can provide useful information about the matrix. It can tell us about properties like invertibility, where a non-zero determinant implies that the matrix is invertible.

For a 3x3 matrix \( \mathbf{A} = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), the determinant is calculated with the formula: \[\det(\mathbf{A}) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Understanding this formula is pivotal. Each term consists of a product and a subtraction from another product, reflecting the combinations of the matrix's elements. This particular format arises from expanding along a row or column, known as cofactor expansion.

Having calculated determinants allows us to solve systems of linear equations, find inverses, and verify properties like \( \det(\mathbf{A}) = \det(\mathbf{A}^T) \), a critical equality in matrix theory. To ensure accuracy, always substitute the correct elements into the determinant formula and carefully perform arithmetic operations.