Problem 35 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(D=\operatorname{diag}\left(a_{1} a_{2} a_{3} \ldots a_{n}\right)\), where \(a_{i} \neq 0\) for all \(i=1\), \(2, \ldots, n\), then \(D^{-1}\) is equal to (A) \(I_{n}\) (B) \(D\) (C) diag \(\left(a_{1}^{-1} a_{2}^{-1} a_{3}^{-1} \ldots a_{n}^{-1}\right)\) (D) None of these

Step-by-Step Solution

Answer

The answer is (C) diag \(\left(a_1^{-1}, a_2^{-1}, a_3^{-1}, \ldots, a_n^{-1}\right)\).

Step 1: Understanding the Problem

We are given a diagonal matrix \(D = \operatorname{diag}(a_1, a_2, a_3, \ldots, a_n)\). We need to find the inverse \(D^{-1}\) of this matrix.

Step 2: Define the Diagonal Matrix

A diagonal matrix \(D\) is a square matrix where all elements outside the main diagonal are zero, and the main diagonal elements are \(a_1, a_2, a_3, \ldots, a_n\). Thus, \(D\) can be represented as:\[D = \begin{pmatrix}a_1 & 0 & 0 & \cdots & 0 \0 & a_2 & 0 & \cdots & 0 \0 & 0 & a_3 & \cdots & 0 \\vdots & \vdots & \vdots & \ddots & \vdots \0 & 0 & 0 & \cdots & a_n\end{pmatrix}\]

Step 3: Finding the Inverse of a Diagonal Matrix

For a diagonal matrix \(D\), its inverse \(D^{-1}\) is also a diagonal matrix where each element on the main diagonal is the reciprocal of the original diagonal elements (since all \(a_i eq 0\)). Thus, \(D^{-1}\) can be written as:\[D^{-1} = \operatorname{diag}\left(a_1^{-1}, a_2^{-1}, a_3^{-1}, \ldots, a_n^{-1}\right)\]

Step 4: Compare with Given Options

Option (C) matches the expression for \(D^{-1}\), which is \(\operatorname{diag}(a_1^{-1}, a_2^{-1}, a_3^{-1}, \ldots, a_n^{-1})\). Hence, \(D^{-1}\) is correctly described by option (C).

Key Concepts

Inverse Matrix

An inverse matrix is crucial in matrix algebra. If a matrix \(A\) has an inverse, it is often denoted as \(A^{-1}\). When a matrix is multiplied by its inverse, the result is the identity matrix, \(I\). This identity matrix acts like the number one in regular arithmetic. The concept of an inverse is applicable only to square matrices, meaning the number of rows and columns must be the same.
However, not all matrices have an inverse. For a matrix to be invertible, or "non-singular", it must have a non-zero determinant. This is like saying a number can only have a reciprocal if it is not zero. In practice, finding the inverse can be tedious for large matrices, but computer algorithms make this manageable.
In the specific case of a diagonal matrix, the inverse is relatively straightforward to find: simply take the reciprocal of each nonzero diagonal element, and reassemble them into a new diagonal matrix.

Diagonal Elements

Diagonal elements in a matrix are those located on the main diagonal, which runs from the top left to the bottom right of a matrix. In a diagonal matrix, all elements off the main diagonal are zero. Because of this, diagonal matrices are simpler to work with in matrix algebra.
For example, consider a diagonal matrix \( D \) with diagonal elements \( a_1, a_2, a_3, \ldots, a_n \). It is represented like this:

• \(D = \begin{pmatrix}a_1 & 0 & \ldots & 0 \ 0 & a_2 & \ldots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \ldots & a_n\end{pmatrix}\)

The simplicity of having zeroes outside the diagonal means certain operations, like finding an inverse, are easier. The inverse of a diagonal matrix, if it exists, will also be diagonal, where each diagonal element is the inverse of the original element. This simplifies many matrix operations significantly.

Matrix Algebra

Matrix algebra is a powerful tool in linear algebra, dealing with matrices and their operations. It includes addition, subtraction, multiplication, and finding inverses. Each operation follows specific rules.
Matrix multiplication is not commutative, meaning \(AB eq BA\) in general. This is different from regular arithmetic. For multiplication, dimensions matter: the number of columns in the first matrix must match the number of rows in the second.
Inverse matrices have a special place in matrix algebra. They allow the solution of linear systems, especially in solving equations of the form \(Ax = b\), commonly found in engineering and computer science applications. Diagonal matrices simplify these operations because they reduce to basic operations on the diagonal elements. Understanding matrix algebra opens doors to understanding systems of equations, transformations, and more.