Problem 16 - Advanced Engineering Mathematics

Question

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{ll} 8 & 0 \\ 0 & \frac{1}{2} \end{array}\right) $$

Step-by-Step Solution

Answer

The inverse of the matrix is \( \begin{pmatrix} \frac{1}{8} & 0 \\ 0 & 2 \end{pmatrix} \).

Step 1: Determining if the Inverse Exists

To find the inverse of a matrix, first, we need to determine if the matrix is invertible. A square matrix is invertible if its determinant is non-zero. So, we calculate the determinant of the given matrix:\[\text{det} \begin{pmatrix} 8 & 0 \ 0 & \frac{1}{2} \end{pmatrix} = (8)(\frac{1}{2}) - (0)(0) = 4\]Since the determinant is 4, which is non-zero, the matrix is invertible.

Step 2: Calculating the Inverse for a Diagonal Matrix

For a diagonal matrix \( \begin{pmatrix} a & 0 \ 0 & d \end{pmatrix} \), the inverse is calculated by taking the reciprocal of the diagonal elements, resulting in:\[\begin{pmatrix} \frac{1}{a} & 0 \ 0 & \frac{1}{d} \end{pmatrix}\]Applying this to the given matrix, we take the reciprocal of 8 and \( \frac{1}{2} \):\[\begin{pmatrix} \frac{1}{8} & 0 \ 0 & 2 \end{pmatrix}\]

Step 3: Verify the Inverse Calculation

To confirm our inverse is correct, we multiply the original matrix by the calculated inverse. The product should result in the identity matrix:\[\begin{pmatrix} 8 & 0 \ 0 & \frac{1}{2} \end{pmatrix} \begin{pmatrix} \frac{1}{8} & 0 \ 0 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]Calculating:\[\begin{pmatrix} (8)(\frac{1}{8}) + 0 & 0 + 0 \ 0 + 0 & (\frac{1}{2})(2) \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]The product is indeed the identity matrix, confirming our inverse is correct.

Key Concepts

Determinant

The determinant is a special number that is calculated from a square matrix. It is crucial to determine whether a matrix is invertible. A matrix must have a non-zero determinant to have an inverse. For a 2x2 matrix, \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is given by the formula:\[ \text{det}(A) = ad - bc \]In our exercise, the determinant of the matrix \( \begin{pmatrix} 8 & 0 \ 0 & \frac{1}{2} \end{pmatrix} \) is calculated as:\[8 \times \frac{1}{2} - 0 \times 0 = 4\]Since the determinant is 4, which is not zero, the matrix is indeed invertible. Knowing how to calculate the determinant is a key skill when working with the inverse of matrices.

Diagonal Matrix

A diagonal matrix is a type of square matrix where all the elements outside the main diagonal are zero. Examples include:- \( \begin{pmatrix} a & 0 \ 0 & d \end{pmatrix} \)These matrices are particularly useful because finding their inverse is straightforward. For a diagonal matrix with non-zero diagonal entries, its inverse is simply another diagonal matrix with reciprocal elements of the original diagonal. More formally:\[\begin{pmatrix} a & 0 \ 0 & d \end{pmatrix}^{-1} = \begin{pmatrix} \frac{1}{a} & 0 \ 0 & \frac{1}{d} \end{pmatrix}\]In the exercise, we apply this to the matrix:\[\begin{pmatrix} 8 & 0 \ 0 & \frac{1}{2} \end{pmatrix}\]delivering the inverse:\[\begin{pmatrix} \frac{1}{8} & 0 \ 0 & 2 \end{pmatrix}\]Diagonal matrices simplify the calculation of inverses, making them easier to work with than non-diagonal matrices.

Identity Matrix

The identity matrix acts as the multiplicative identity in matrix algebra, similar to how the number 1 works in multiplication of numbers. It is a square matrix with ones on the main diagonal and zeros elsewhere:- \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \)Multiplying any matrix by the identity matrix results in the original matrix. This is a key property because when you multiply a matrix by its inverse, the product should be the identity matrix. This verifies that the calculated inverse is correct.In the exercise:\[\begin{pmatrix} 8 & 0 \ 0 & \frac{1}{2} \end{pmatrix} \times \begin{pmatrix} \frac{1}{8} & 0 \ 0 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]The product results in the identity matrix, confirming that our inverse calculation was accurate. Understanding the identity matrix is essential when working with inverses, as it provides immediate feedback on the correctness of your computations.