Problem 19 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Both determinants are equal: \( \det \mathbf{A} = -2 \) and \( \det \mathbf{A}^{T} = -2 \).

Step 1: Compute the Determinant of Matrix A

To find \( \det \mathbf{A} \), we will use the formula for the determinant of a 3x3 matrix: \[ \det \mathbf{A} = a(ei - fh) - b(di - fg) + c(dh - eg) \]For the given matrix \( \mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \ 4 & 1 & -1 \ 1 & 2 & -1 \end{pmatrix} \), we identify the elements:\( a = 1, b = 2, c = 1, d = 4, e = 1, f = -1, g = 1, h = 2, i = -1 \).Plugging in these values, the determinant is:\[ \det \mathbf{A} = 1((1)(-1) - (-1)(2)) - 2((4)(-1)-(1)(1)) + 1((4)(2)-(1)(1)) \] \[ = 1(-1 + 2) - 2(-4 - 1) + 1(8 - 1) \] \[ = 1 - 10 + 7 \] \[ = -2 \].

Step 2: Transpose the Matrix A

The transpose of matrix \( \mathbf{A} \) is obtained by swapping rows and columns. Thus, the transpose \( \mathbf{A}^{T} \) is \[ \mathbf{A}^{T} = \begin{pmatrix} 1 & 4 & 1 \ 2 & 1 & 2 \ 1 & -1 & -1 \end{pmatrix} \].

Step 3: Compute the Determinant of Transposed Matrix A

Using the same formula for the determinant:\[ \det \mathbf{A}^{T} = a(ei - fh) - b(di - fg) + c(dh - eg) \]For \( \mathbf{A}^{T} = \begin{pmatrix} 1 & 4 & 1 \ 2 & 1 & 2 \ 1 & -1 & -1 \end{pmatrix} \), we identify the elements:\( a = 1, b = 4, c = 1, d = 2, e = 1, f = 2, g = 1, h = -1, i = -1 \).Plugging in these values, the determinant is:\[ \det \mathbf{A}^{T} = 1(1(-1) - 2(-1)) - 4(2(-1) - 1(1)) + 1(2(-1) - 1(1)) \] \[ = 1(-1 + 2) + 4(2 - 1) + 1(-2 -1) \] \[ = 1 - 10 + 7 \] \[ = -2 \].

Step 4: Verify Equality of Determinants

We found that \( \det \mathbf{A} = -2 \) and \( \det \mathbf{A}^{T} = -2 \). Thus, the determinants of the matrix \( \mathbf{A} \) and its transpose \( \mathbf{A}^{T} \) are indeed equal, verifying that \( \det \mathbf{A} = \det \mathbf{A}^{T} \).

Key Concepts

Matrix Transpose

The transpose of a matrix involves flipping its rows and columns. Think of it as rotating the matrix across its diagonal. If you have a matrix \( \mathbf{A} \), the transpose, denoted as \( \mathbf{A}^{T} \), is constructed by turning the first row of \( \mathbf{A} \) into the first column of \( \mathbf{A}^{T} \), the second row into the second column, and so forth.

For example, given the matrix \( \mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \ 4 & 1 & -1 \ 1 & 2 & -1 \end{pmatrix} \), its transpose is \( \mathbf{A}^{T} = \begin{pmatrix} 1 & 4 & 1 \ 2 & 1 & 2 \ 1 & -1 & -1 \end{pmatrix} \). Notice how each element in the new matrix corresponds to a flipped position from the original.

Why is transposing important? It's crucial in many matrix operations and properties, including calculating determinants. The transpose can change the orientation of data and is often used in areas like data analysis and computer graphics.

3x3 Matrix Determinant Formula

The determinant of a 3x3 matrix is a special number that provides insights into matrix properties, such as whether a matrix is invertible, or the volume ratio of a transformed space. For a 3x3 matrix \( \mathbf{A} \), it is calculated using the formula:

\[\det \mathbf{A} = a(ei - fh) - b(di - fg) + c(dh - eg)\]

This may look complicated, but let's break it down into steps:

• Identify the components of the matrix, where each letter represents a position in a matrix, like rows and columns.
• Substitute these elements into the formula.
• Perform the arithmetic operations to compute the determinant.

Using this formula, you can also cross-verify your answers or apply it in various applications such as solving systems of linear equations or analyzing geometric transformations.

Properties of Determinants

Determinants have interesting properties that help simplify calculations and reveal characteristics about matrices. Understanding these properties can be very helpful when dealing with matrices:

• Equality with Transpose: The determinant of a matrix is the same as the determinant of its transpose. Mathematically, \( \det \mathbf{A} = \det \mathbf{A}^{T} \). This means you can transpose matrices when it simplifies your calculations without affecting the determinant.
• Determinant Zero: If a matrix's determinant is zero, the matrix is singular, meaning it doesn't have an inverse.
• Multiplicative Property: The determinant of a product of matrices equals the product of the determinants, i.e., \( \det(\mathbf{AB}) = \det(\mathbf{A}) \times \det(\mathbf{B}) \).
• Row/Column Operations: Swapping two rows or columns changes the sign of the determinant. If two rows or columns are identical, the determinant is zero.

These properties make determinants a powerful tool in linear algebra, aiding in computations and providing insights into the nature of matrix transformations.