Problem 247 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Show that the inverse of a diagonal matrix is a diagonal matrix.

Step-by-Step Solution

Answer

The inverse of a diagonal matrix 'D', with diagonal elements \(d_1, d_2, ..., d_n\), is a diagonal matrix with elements \(1/d_1, 1/d_2, ..., 1/d_n\) on its diagonal, respectively, assuming that no element on the diagonal of original matrix is zero.

Step 1: Understand diagonal matrices

Diagonal matrices are special types of square matrices that have all their elements zero, except the ones present diagonally from top left to bottom right. Suppose we have a matrix 'D' with elements \(d_1, d_2, ..., d_n\) on its diagonal.

Step 2: Define inverse of a matrix

The inverse of a matrix 'A' is denoted by \(A^{-1}\) and it is that matrix which when multiplied by 'A' gives the identity matrix 'I' as the product i.e. \(AA^{-1} = I\). Our task is to find this \(A^{-1}\) for our diagonal matrix 'D'.

Step 3: Formulate the inverse of a diagonal matrix

For a diagonal matrix 'D', with diagonal elements \(d_1, d_2, ..., d_n\), the inverse will have elements \(1/d_1, 1/d_2, ..., 1/d_n\) on its diagonal, respectively, given no element on the diagonal of original matrix is zero. This is because multiplying \(d_i * 1/d_i\) for each 'i' gives us 1, which is the element on the identity matrix, leaving rest of the elements as zero.

Step 4: Verify the result

We can ascertain this by multiplying the diagonal matrix 'D' with its inverse 'D^-1'. The result is indeed the identity matrix 'I', confirming that the inverse of a diagonal matrix is also a diagonal matrix.

Key Concepts

Diagonal Matrices

Imagine a checkerboard where all the checkers are aligned perfectly in a row from one corner to the other. A diagonal matrix is somewhat like that checkerboard – it's a special kind of square matrix where all the elements are zero except those along the 'checker line', which is the main diagonal running from the top-left corner to the bottom-right corner.

Let's say we have a diagonal matrix named 'D'. It looks like this:
\begin{align*}D = \begin{pmatrix}d_1 & 0 & \cdots & 0 \0 & d_2 & \cdots & 0 \vdots & \vdots & \ddots & \vdots \0 & 0 & \cdots & d_n \end{pmatrix}\begin{align*}where the numbers \(d_1, d_2, ..., d_n\) are sitting on that 'checker line'. This simplicity makes diagonal matrices quite friendly for all sorts of mathematical operations, like finding inverses, as you'll soon see!

Matrix Inverse

Now, let's chat about what it means for a matrix to have an inverse. Imagine you have a magical ticket that takes you to a concert, and you're having the time of your life. The inverse of that ticket would be the one that, when used, takes you back home instantly, returning everything to normal – no concert, just your usual environment.

Similarly, the inverse of a matrix, denoted by \(A^{-1}\), takes the matrix 'A' back to the mathematical equivalent of 'normal', which is the identity matrix 'I'. This is like pressing the reset button on a calculator. The inverse does this via the matrix multiplication dance – when you multiply 'A' by \(A^{-1}\), you get the identity matrix as a result, just as when you used the magical homecoming ticket: \(A A^{-1} = I\).It's pretty magical, except in mathematics, we don't rely on magic; we prove things step by step, just like how we'd strategically plan our escape from a concert if we really needed to.

Identity Matrix

Ever played the 'Spot the Difference' game? Well, the identity matrix is like that picture in the game where there are no differences to spot – it’s perfection! An identity matrix, denoted as 'I', can be any size as long as it's square, and it has 1's all down the main diagonal (our 'checker line') and 0's everywhere else.

\begin{align*}I_n = \begin{pmatrix}1 & 0 & \cdots & 0 \0 & 1 & \cdots & 0 \vdots & \vdots & \ddots & \vdots \0 & 0 & \cdots & 1 \end{pmatrix}\begin{align*}Think of it as the neutral element in matrix multiplication; multiply any matrix 'A' by 'I', and 'A' will look right back at you unaltered: \(A I = A\). It's like the echo in the canyon that perfectly mimics your words. The identity matrix maintains everything as is, affirming its status as the ultimate matrix wallflower – it's there, but it doesn't change the scene.