Problem 1 - Advanced Engineering Mathematics

Question

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

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

Step 1: Understanding Matrix Inversion

The product of a matrix and its inverse is the identity matrix. This means that if \( \mathbf{A} \mathbf{B} = \mathbf{I} \), where \( \mathbf{I} \) is the identity matrix, then \( \mathbf{B} \) is the inverse of \( \mathbf{A} \). For a 2x2 matrix, the identity matrix is \( \mathbf{I} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \).

Step 2: Multiply Matrices \( \mathbf{A} \) and \( \mathbf{B} \)

Multiply the matrices \( \mathbf{A} \) and \( \mathbf{B} \) to check if their product is the identity matrix. Calculate each element of the resulting matrix using the multiplication rule for matrices: \((AB)_{ij} = \sum_k A_{ik}B_{kj}\).Given \( \mathbf{A} = \begin{pmatrix} 1 & \frac{1}{2} \ 2 & \frac{3}{2} \end{pmatrix} \) and \( \mathbf{B} = \begin{pmatrix} 3 & -1 \ -4 & 2 \end{pmatrix} \):

Step 3: Calculate Product Element \((1,1)\)

To find the element in the first row, first column of \( \mathbf{AB} \), use:\[ (AB)_{11} = 1 \cdot 3 + \frac{1}{2} \cdot (-4) = 3 - 2 = 1 \]

Step 4: Calculate Product Element \((1,2)\)

To find the element in the first row, second column of \( \mathbf{AB} \), use:\[ (AB)_{12} = 1 \cdot (-1) + \frac{1}{2} \cdot 2 = -1 + 1 = 0 \]

Step 5: Calculate Product Element \((2,1)\)

To find the element in the second row, first column of \( \mathbf{AB} \), use:\[ (AB)_{21} = 2 \cdot 3 + \frac{3}{2} \cdot (-4) = 6 - 6 = 0 \]

Step 6: Calculate Product Element \((2,2)\)

To find the element in the second row, second column of \( \mathbf{AB} \), use:\[ (AB)_{22} = 2 \cdot (-1) + \frac{3}{2} \cdot 2 = -2 + 3 = 1 \]

Step 7: Check Resulting Matrix

After calculations, the product matrix \( \mathbf{AB} \) is:\[ \mathbf{AB} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]This is the identity matrix, confirming that \( \mathbf{B} \) is the inverse of \( \mathbf{A} \).

Key Concepts

Identity Matrix

An identity matrix is a special kind of matrix that acts as a multiplicative identity for matrices. It's like the role of the number 1 in regular multiplication. This matrix can be square, meaning it has the same number of rows and columns, which is a key condition.
For a 2x2 matrix, the identity matrix is \( \mathbf{I} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \). When you multiply any 2x2 matrix with this identity matrix, the original matrix remains unchanged. It's like a magic trick, where the matrix essentially stays the same.
This property is crucial when dealing with matrix inverses. When you multiply a matrix by its inverse, you get the identity matrix as the result. This is your way to verify that one matrix really is the inverse of another.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, but it's not as straightforward as multiplying regular numbers. When multiplying two matrices, say \( \mathbf{A} \) and \( \mathbf{B} \), each element in the resulting matrix is calculated by multiplying elements from the rows of \( \mathbf{A} \) with corresponding elements from the columns of \( \mathbf{B} \), and then adding them up.
For example, if you want to find the element in the first row, first column of the product matrix \( \mathbf{C} = \mathbf{AB} \), you use the formula:

• \( (AB)_{11} = A_{11} \times B_{11} + A_{12} \times B_{21} \)

It's important to note that matrix multiplication is not commutative; that means \( \mathbf{A} \times \mathbf{B} \) is not necessarily the same as \( \mathbf{B} \times \mathbf{A} \).
This operation allows us to check if two matrices, such as \( \mathbf{A} \) and \( \mathbf{B} \), are inverses of each other. If the product \( \mathbf{AB} \) yields the identity matrix, then \( \mathbf{B} \) is indeed the inverse of \( \mathbf{A} \).

Inverse of a Matrix

The inverse of a matrix is akin to the reciprocal of a number. When you multiply a matrix by its inverse, you get the identity matrix. Not every matrix has an inverse, but when a matrix does have one, it's called an invertible or non-singular matrix.
The inverse of a 2x2 matrix \( \mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) can be found using the formula:
\[ \mathbf{A}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]
Here, \( ad-bc \) must not be zero, as this would make the matrix non-invertible. This term is known as the determinant of the matrix.
Finding the inverse is especially useful because it allows solution of systems of linear equations, among other applications. In the exercise provided, matrix \( \mathbf{B} \) is shown to be the inverse of matrix \( \mathbf{A} \) because their product results in the identity matrix.