Problem 30 - Advanced Engineering Mathematics

Question

If \(\mathbf{A}\) is nonsingular, then \(\left(\mathbf{A}^{T}\right)^{-1}=\left(\mathbf{A}^{-1}\right)^{T}\). Verify this for \(\mathbf{A}=\left(\begin{array}{rr}1 & 4 \\ 2 & 10\end{array}\right) .\)

Step-by-Step Solution

Answer

The property \( \left(\mathbf{A}^{T}\right)^{-1} = \left(\mathbf{A}^{-1}\right)^{T} \) is verified for the given matrix \( \mathbf{A} \).

Step 1: Calculate the Inverse of Matrix A

Given matrix \( \mathbf{A} = \begin{pmatrix} 1 & 4 \ 2 & 10 \end{pmatrix} \), we need to calculate its inverse. The formula for the inverse of a 2x2 matrix \( \mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is \( \mathbf{A}^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix} \). Compute the determinant: \( (1)(10) - (4)(2) = 10 - 8 = 2 \). Thus, \( \mathbf{A}^{-1} = \frac{1}{2} \begin{pmatrix} 10 & -4 \ -2 & 1 \end{pmatrix} \).

Step 2: Transpose the Inverse of Matrix A

From Step 1, we have \( \mathbf{A}^{-1} = \frac{1}{2} \begin{pmatrix} 10 & -4 \ -2 & 1 \end{pmatrix} \). To find the transpose, swap its rows and columns: \( \left(\mathbf{A}^{-1}\right)^{T} = \frac{1}{2} \begin{pmatrix} 10 & -2 \ -4 & 1 \end{pmatrix} \).

Step 3: Calculate the Transpose of Matrix A

Find the transpose of matrix \( \mathbf{A} \). Given \( \mathbf{A} = \begin{pmatrix} 1 & 4 \ 2 & 10 \end{pmatrix} \), its transpose is \( \mathbf{A}^{T} = \begin{pmatrix} 1 & 2 \ 4 & 10 \end{pmatrix} \).

Step 4: Find the Inverse of the Transpose of Matrix A

Use the formula for the inverse of a 2x2 matrix again for \( \mathbf{A}^{T} = \begin{pmatrix} 1 & 2 \ 4 & 10 \end{pmatrix} \). Compute the determinant: \( (1)(10) - (2)(4) = 10 - 8 = 2 \). The inverse is \( \left(\mathbf{A}^{T}\right)^{-1} = \frac{1}{2} \begin{pmatrix} 10 & -2 \ -4 & 1 \end{pmatrix} \).

Step 5: Verify the Property

Compare \( \left(\mathbf{A}^{T}\right)^{-1} = \frac{1}{2} \begin{pmatrix} 10 & -2 \ -4 & 1 \end{pmatrix} \) and \( \left(\mathbf{A}^{-1}\right)^{T} = \frac{1}{2} \begin{pmatrix} 10 & -2 \ -4 & 1 \end{pmatrix} \). They are identical, verifying the property \( \left(\mathbf{A}^{T}\right)^{-1} = \left(\mathbf{A}^{-1}\right)^{T} \).

Key Concepts

Transpose of a Matrix

Transposing a matrix involves swapping its rows and columns. This essentially means that the first row of the original matrix becomes the first column of the new matrix, the second row becomes the second column, and so on. For example, if you have a matrix \( \mathbf{A} = \begin{pmatrix} 1 & 4 \ 2 & 10 \end{pmatrix} \), its transpose would be \( \mathbf{A}^{T} = \begin{pmatrix} 1 & 2 \ 4 & 10 \end{pmatrix} \).

• Transposing is a commonly used operation in matrix algebra due to its simple nature and useful properties.
• One important property of the transpose is that the transpose of a transpose will return the original matrix: \( (\mathbf{A}^{T})^{T} = \mathbf{A} \).

The transpose operation preserves certain characteristics of a matrix, such as symmetry. If a matrix is symmetrical (i.e., its transpose equals itself), it simplifies various calculations and theories in linear algebra.

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix like \( \mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated by the formula \( ad - bc \).
Determination of the determinant is crucial in matrix algebra because it provides essential information about the matrix, such as:

• If the determinant is zero, the matrix is singular, meaning it does not have an inverse.
• If the determinant is non-zero, the matrix is nonsingular and has an inverse.

In our scenario, for \( \mathbf{A} = \begin{pmatrix} 1 & 4 \ 2 & 10 \end{pmatrix} \), the determinant is\( 1 \times 10 - 4 \times 2 = 10 - 8 = 2 \).
This non-zero determinant indicates that the matrix is invertible, which is important for solving linear equations and other pivotal operations in linear algebra.

Matrix Properties

Matrices possess various useful properties that simplify matrix operations and problem-solvingin algebra. Understanding these properties helps in employing matrices effectively. Let's look at some core properties, including those useful for matrix inversion:

• Associative Property: For any matrices \( \mathbf{A}, \mathbf{B}, \mathbf{C} \) of commensurate dimensions, \( (\mathbf{A} \cdot \mathbf{B}) \cdot \mathbf{C} = \mathbf{A} \cdot (\mathbf{B} \cdot \mathbf{C}) \).
• Distributive Property: Matrix multiplication distributes over addition: \( \mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C} \).
• Identical Property: Multiplying any matrix by an identity matrix of corresponding size results in the original matrix: \( \mathbf{A} \cdot \mathbf{I} = \mathbf{A} \).
• Inverse Property: If \( \mathbf{A} \) is nonsingular, then \( \mathbf{A}^{-1} \cdot \mathbf{A} = \mathbf{I} \), the identity matrix.

These properties are fundamental in validating the expression \( (\mathbf{A}^{T})^{-1} = (\mathbf{A}^{-1})^{T} \) as seen in matrix inversion problems, enabling simpler checks and verifications on matrix operations.