Problem 40 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The inverse matrix 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: Understand the Problem

We need to find the inverse of a diagonal matrix \(A\): \[ A = \begin{bmatrix} a & 0 & 0 & 0 \ 0 & b & 0 & 0 \ 0 & 0 & c & 0 \ 0 & 0 & 0 & d \end{bmatrix} \]We are given that \(a, b, c,\) and \(d\) are non-zero, which ensures the matrix has an inverse.

Step 2: Recall Inverse of a Diagonal Matrix

The inverse of a diagonal matrix is another diagonal matrix where each diagonal element is replaced by its reciprocal. If \(A = \begin{bmatrix} a & 0 & 0 & 0 \ 0 & b & 0 & 0 \ 0 & 0 & c & 0 \ 0 & 0 & 0 & d \end{bmatrix}\), then \(A^{-1} = \begin{bmatrix} \frac{1}{a} & 0 & 0 & 0 \ 0 & \frac{1}{b} & 0 & 0 \ 0 & 0 & \frac{1}{c} & 0 \ 0 & 0 & 0 & \frac{1}{d} \end{bmatrix}\), given each diagonal entry is non-zero.

Step 3: Write Down the Inverse Matrix

Using the rule from Step 2, the inverse of the given diagonal matrix \(A\) is: \[ A^{-1} = \begin{bmatrix} \frac{1}{a} & 0 & 0 & 0 \ 0 & \frac{1}{b} & 0 & 0 \ 0 & 0 & \frac{1}{c} & 0 \ 0 & 0 & 0 & \frac{1}{d} \end{bmatrix} \]

Key Concepts

Diagonal Matrix

In the world of matrices, a diagonal matrix is a special type of matrix where all the non-diagonal elements are zero. Imagine a large square grid filled with numbers. In a diagonal matrix, the only numbers that aren't zero are those that run from the top left to the bottom right corner — this line is called the "diagonal." These non-zero elements can be any number, provided the elements outside this diagonal line are strictly zero. This matrix's unique structure allows us to perform operations like matrix inversions very easily.

Key characteristics of a diagonal matrix include:

• It is always a square matrix, meaning it has as many rows as columns.
• The diagonal elements can be non-zero numbers.
• The simplicity in structure leads to significant computational advantages, especially in finding the inverse.

In our exercise, the matrix is a 4x4 diagonal matrix with non-zero diagonal elements \(a, b, c,\) and \(d\), ensuring that the matrix can be inverted.

Matrix Inversion

Matrix inversion is the process of finding another matrix, which when multiplied with the original, returns the identity matrix. This concept is analogous to finding the reciprocal of a number — where a number \(x\) times its reciprocal \(1/x\) equals 1. Similarly, a matrix \(A\) multiplied by its inverse \(A^{-1}\) yields the identity matrix, denoted as \(I\).

Some points regarding matrix inversion:

• Not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.
• Inversion can be computationally intensive for large matrices, but diagonal matrices offer a much simpler route to inversion.
• For our diagonal matrix, each diagonal element is easily inverted by taking its reciprocal.

In the context of the exercise, because the matrix is diagonal, its inversion is straightforward. The inverse is a diagonal matrix with the reciprocal of each original diagonal element, turning \(a, b, c,\) and \(d\) into \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c},\) and \(\frac{1}{d}\).

Reciprocal Elements

The concept of reciprocal elements is simple yet powerful, especially in matrix mathematics. A reciprocal of a number \(x\) is \(1/x\); when multiplied together, these values yield 1. In matrices, finding the inverse often involves using reciprocals, particularly with diagonal matrices.

Here’s how reciprocals work in matrix inversion:

• For a diagonal matrix, inversing each diagonal element results in the matrix's inverse, as these are the only non-zero numbers.
• This means taking each diagonal value and finding its reciprocal results in the inverse matrix. For example, if the original diagonal has values \(a, b, c,\) and \(d\), the inverse diagonal will have \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c},\) and \(\frac{1}{d}\).

The exercise highlights the efficiency of using reciprocal elements to invert a diagonal matrix, simplifying an otherwise complex operation into basic arithmetic.