Problem 53 - Algebra for JEE ADVANCED

Question

The inverse of a diagonal matrix is a. a diagonal matrix b. a skew-symmetric matrix c. a symmetric matrix d. none of these

Step-by-Step Solution

Answer

The inverse of a diagonal matrix is a diagonal matrix.

Step 1: Understanding a Diagonal Matrix

A diagonal matrix is a square matrix where all elements outside the main diagonal are zero. Thus, if the matrix is denoted as \( D \), then \( D(i, j) = 0 \) for all \( i eq j \). The main diagonal can have non-zero elements.

Step 2: Inverse of a Diagonal Matrix

The inverse of a diagonal matrix is also a diagonal matrix. For a diagonal matrix \( D \), if it is invertible, its elements \( d_{ii} \) on the diagonal are non-zero, and its inverse will be a matrix \( D^{-1} \) where the elements \( d^{-1}_{ii} = 1/d_{ii} \). All other elements remain zero.

Step 3: Selecting the Correct Option

From the above properties of diagonal matrices, we see the inverse remains a diagonal matrix since all off-diagonal elements remain zero and the inverse of each diagonal entry is simply the reciprocal of the original entry. Therefore, the correct option is **a. a diagonal matrix**.

Key Concepts

Understanding Diagonal Matrices

A diagonal matrix is a special kind of matrix. What's unique about it is that all the elements found off the main diagonal are zero.
This means, if the matrix is represented by the letter \( D \), then every element \( D(i, j) \) is zero for all \( i eq j \). The main diagonal can contain any numbers, usually non-zero.
This makes diagonal matrices particularly simple since they are defined by just the main diagonal values.

• Example: \( \begin{bmatrix} 3 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 7 \end{bmatrix} \)
• In this example, only 3, 5, and 7 on the diagonal are non-zero, the rest are zeroes.

Diagonal matrices are easier to work with in mathematics due to their simplicity.

Matrix Inversion Explained

Matrix inversion is the process of finding a matrix that, when multiplied by the original matrix, results in what's called the identity matrix.
The identity matrix acts like the number 1 for matrices, leaving other matrices unchanged when multiplied. For diagonal matrices, this inversion is especially straightforward.
For a diagonal matrix \( D \) to have an inverse, each element on its diagonal must be non-zero. If \( D \) is invertible, then its inverse \( D^{-1} \) is also diagonal.

• Example: If \( D = \begin{bmatrix} a & 0 \ 0 & b \end{bmatrix} \), then \( D^{-1} = \begin{bmatrix} 1/a & 0 \ 0 & 1/b \end{bmatrix} \)
• All off-diagonal elements remain zero, simplifying calculations.

This characteristic makes them easy to manage, as calculating their inverse involves just finding the reciprocal of their diagonal elements.

Properties of Matrices in Focus

Understanding the properties of matrices can make solving problems much more intuitive. Diagonal matrices come with their own set of beneficial properties.
Here's why they are convenient:

• Diagonal matrices are easy to multiply. The product of two diagonal matrices is another diagonal matrix where each diagonal element is the product of the respective diagonal elements.
• They're also simple to raise to a power. To do so, raise each diagonal element to that power.

These properties relate to the simplicity with which diagonal matrices can be inverted, as each non-zero element is inverted while retaining the zeroes elsewhere.
The straightforward nature of diagonal matrices makes them a fundamental part of linear algebra.