Problem 42 - Algebra and Trigonometry

Question

Find the inverse of the matrix. $$ \begin{array}{l}{\left[\begin{array}{cccc}{a} & {0} & {0} & {0} \\ {0} & {b} & {0} & {0} \\ {0} & {0} & {c} & {0} \\ {0} & {0} & {0} & {d} \\\ {}\end{array}\right]} \\ {(a b c d \neq 0)}\end{array} $$

Step-by-Step Solution

Answer

The inverse is \(\begin{bmatrix} \frac{1}{a} & 0 & 0 & 0 \\ 0 & \frac{1}{b} & 0 & 0 \\ 0 & 0 & \frac{1}{c} & 0 \\ 0 & 0 & 0 & \frac{1}{d} \end{bmatrix}\).

Step 1: Understanding the Problem

You are asked to find the inverse of a diagonal matrix. Diagonal matrices have non-zero elements only on their main diagonal, and all off-diagonal elements are zero. For the given matrix, the diagonal elements are \(a, b, c,\) and \(d\), all of which are non-zero, ensuring that the matrix is invertible.

Step 2: Formula for Inverse of a Diagonal Matrix

The inverse of a diagonal matrix is another diagonal matrix where each diagonal element is the reciprocal of the corresponding element in the original matrix. This means if the original matrix is \(\begin{bmatrix} a & 0 & 0 & 0 \ 0 & b & 0 & 0 \ 0 & 0 & c & 0 \ 0 & 0 & 0 & d \end{bmatrix}\), then the inverse will be \(\begin{bmatrix} \frac{1}{a} & 0 & 0 & 0 \ 0 & \frac{1}{b} & 0 & 0 \ 0 & 0 & \frac{1}{c} & 0 \ 0 & 0 & 0 & \frac{1}{d} \end{bmatrix}\).

Step 3: Calculating the Inverse Matrix

Using the formula for the inverse of a diagonal matrix from Step 2, construct the inverse matrix by taking the reciprocal of each diagonal element. Thus, the resulting inverse matrix becomes: \[\begin{bmatrix}\frac{1}{a} & 0 & 0 & 0 \0 & \frac{1}{b} & 0 & 0 \0 & 0 & \frac{1}{c} & 0 \0 & 0 & 0 & \frac{1}{d} \end{bmatrix}\]

Step 4: Verify the Solution

To ensure that this is the correct inverse, multiply the original matrix by the calculated inverse. The result should be the identity matrix:\[\begin{bmatrix} a & 0 & 0 & 0 \ 0 & b & 0 & 0 \ 0 & 0 & c & 0 \ 0 & 0 & 0 & d \end{bmatrix} \times \begin{bmatrix}\frac{1}{a} & 0 & 0 & 0 \0 & \frac{1}{b} & 0 & 0 \0 & 0 & \frac{1}{c} & 0 \0 & 0 & 0 & \frac{1}{d} \end{bmatrix} =\begin{bmatrix}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1 \end{bmatrix}\]This multiplication confirms the correctly found inverse.

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all the entries outside the main diagonal are zero. This means that only the elements down the main diagonal can be non-zero, simplifying many matrix operations. Diagonal matrices are significant because they make complex computations easier since you only focus on the diagonal elements.

• Property: If a matrix is diagonal, it is square, but not every square matrix is diagonal.
• Example: In a 4x4 matrix, a diagonal matrix would look like: \[ \begin{bmatrix} a & 0 & 0 & 0 \ 0 & b & 0 & 0 \ 0 & 0 & c & 0 \ 0 & 0 & 0 & d \end{bmatrix} \]
• Applications: Diagonal matrices are used in simplifying problems like finding eigenvalues and solving systems of equations.

When finding inverses, diagonal matrices remain diagonal, making calculations straightforward.

Matrix Inversion

Matrix inversion is the process of finding another matrix that, when multiplied with the original matrix, results in an identity matrix. This other matrix is known as the inverse matrix.

• Condition: A matrix must be square and have a non-zero determinant to have an inverse.
• Importance: Finding an inverse is essential in solving linear systems and various applications in engineering and computer science.
• Diagonal Matrices: For diagonal matrices, finding the inverse is simple since the inverse of a diagonal matrix is another diagonal matrix with the reciprocals of the original diagonal elements.

Remember, not all matrices have inverses, but for those that do, the inverse can solve equations like \(Ax = b\) by transforming it to \(x = A^{-1}b\).

Identity Matrix

The identity matrix is a unique square matrix that acts as the multiplicative identity in matrix algebra. Essentially, multiplying any matrix by an identity matrix returns the original matrix, similar to how multiplying any number by 1 leaves it unchanged.

• Notation: It is often denoted as \(I_n\), where \(n\) is the size of the matrix, like \(I_4\) for a 4x4 identity matrix.
• Structure: An identity matrix has ones on the main diagonal and zeros elsewhere, such as: \[ \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix} \]
• Role in Inversion: In matrix inversion, multiplying a matrix by its inverse results in the identity matrix, confirming that the solution is correct.

The identity matrix is crucial in mathematical expressions and operations, serving as a foundational element in linear algebra.