Problem 1 - Advanced Engineering Mathematics

Question

In Problems 1 and 2 , verify that the matrix \(\mathbf{B}\) is the inverse of the matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{ll} 1 & \frac{1}{2} \\ 2 & \frac{3}{2} \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr} 3 & -1 \\ -4 & 2 \end{array}\right) $$

Step-by-Step Solution

Answer

\( \mathbf{B} \) is the inverse of \( \mathbf{A} \) as their product is the identity matrix.

Step 1: Understand the Problem

We need to verify that matrix \( \mathbf{B} \) is the inverse of matrix \( \mathbf{A} \). This means that when \( \mathbf{A} \) is multiplied by \( \mathbf{B} \), it should result in the identity matrix \( \mathbf{I} \).

Step 2: Set Up the Matrix Multiplication

To verify the inverse, calculate \( \mathbf{A} \times \mathbf{B} \). The matrices are:\[ \mathbf{A} = \begin{pmatrix} 1 & \frac{1}{2} \ 2 & \frac{3}{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 3 & -1 \ -4 & 2 \end{pmatrix} \]Multiply these matrices to get \( \mathbf{A} \times \mathbf{B} \).

Step 3: Perform Matrix Multiplication

To perform the multiplication, calculate each element as follows:- For element \( (1,1) \): \( 1 \times 3 + \frac{1}{2} \times (-4) = 3 - 2 = 1 \)- For element \( (1,2) \): \( 1 \times (-1) + \frac{1}{2} \times 2 = -1 + 1 = 0 \)- For element \( (2,1) \): \( 2 \times 3 + \frac{3}{2} \times (-4) = 6 - 6 = 0 \)- For element \( (2,2) \): \( 2 \times (-1) + \frac{3}{2} \times 2 = -2 + 3 = 1 \)

Step 4: Verify the Result

The resulting matrix from \( \mathbf{A} \times \mathbf{B} \) is:\[ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]This is the 2x2 identity matrix, confirming that \( \mathbf{B} \) is indeed the inverse of \( \mathbf{A} \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are multiplied to produce a third matrix. This operation combines elements from the rows of the first matrix with the columns of the second matrix in a specific way.
For example, if you have a matrix \( \mathbf{A} \) represented as:

• Rows: \( a_{11}, a_{12} \) and \( a_{21}, a_{22} \)
• Columns are those same pairs flipped when considering the next matrix \( \mathbf{B} \).

To multiply \( \mathbf{A} \) by another matrix \( \mathbf{B} \), you perform operations on each element as follows:

• The element in the first row, first column of the product matrix results from the sum of products of the elements of the first row of the first matrix and the first column of the second matrix.
• This pattern continues for all positions in the resulting matrix, ensuring all elements relate to corresponding row and column operations.

Matrix multiplication is not commutative, meaning \( \mathbf{A} \times \mathbf{B} eq \mathbf{B} \times \mathbf{A} \), so the order of multiplication matters greatly.

Identity Matrix

The identity matrix is an essential concept in matrix algebra. It functions similarly to the number 1 in regular multiplication, meaning any matrix multiplied by an identity matrix remains unchanged. In the case of 2x2 matrices, the identity matrix looks like:

• \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \)

When verifying inverse matrices, the goal is to multiply a matrix by its supposed inverse and obtain this identity matrix as the result.
This unique characteristic serves as a proof of inverses in matrix operations. If matrix \( \mathbf{A} \) and \( \mathbf{B} \) are inverses, then both \( \mathbf{A} \times \mathbf{B} = \mathbf{I} \) and \( \mathbf{B} \times \mathbf{A} = \mathbf{I} \) hold true.
Recognizing the identity matrix helps confirm that matrix operations have been correctly performed and that an inverse matrix has been correctly identified.

2x2 Matrices

2x2 matrices are a simple yet powerful tool in linear algebra, commonly used for practical applications like geometry and system transformations.
A 2x2 matrix is usually represented as:

• \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

These matrices conveniently model linear transformations and provide a manageable framework for understanding more complex matrix concepts. To determine the inverse of a 2x2 matrix \( \mathbf{A} \), such that \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), one finds:

• Use the formula \( A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \)
• \( ad - bc \) is called the determinant and must be non-zero.

The inverse only exists if the determinant is not zero, ensuring the matrix is invertible. Mastery of 2x2 matrices allows for quick calculations and applications, forming the backbone for larger, more complex matrix problems.