Problem 35 - A Complete Resource Book in Mathematics for JEE Main 2017

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 \(\left(\begin{array}{lll}a_{1}^{-1} & a_{2}^{-1} & a_{3}^{-1} & \ldots & a_{n}^{-1}\end{array}\right)\) (D) None of these

Step-by-Step Solution

Answer

The inverse \( D^{-1} \) is option (C).

Step 1: Understand the Problem

We are given a diagonal matrix \( D = \operatorname{diag}(a_1, a_2, a_3, \ldots, a_n) \) where each \( a_i eq 0 \). Our task is to determine the inverse of this matrix \( D^{-1} \).

Step 2: Recall Properties of Diagonal Matrices

A diagonal matrix has non-zero elements only on its main diagonal. For a diagonal matrix, the inverse is computed by taking the reciprocal of each non-zero element on the diagonal, provided all diagonal entries are non-zero.

Step 3: Formulate Expression for Inverse

Based on the properties of diagonal matrices, the inverse \( D^{-1} \) can be expressed as another diagonal matrix where each diagonal element from the original matrix \( D \) is replaced by its reciprocal. Thus, \( D^{-1} = \operatorname{diag}(a_1^{-1}, a_2^{-1}, a_3^{-1}, \ldots, a_n^{-1}) \).

Step 4: Compare Options

The problem asks us to choose between options for \( D^{-1} \). Given our formulation, \( D^{-1} = \operatorname{diag}(a_1^{-1}, a_2^{-1}, a_3^{-1}, \ldots, a_n^{-1}) \), which matches choice (C).

Step 5: Conclude the Solution

Since \( D^{-1} \) matches option (C), it is the correct answer. The valid inverse for \( D \) is indeed the diagonal matrix formed by the reciprocals of the diagonal elements of \( D \).

Key Concepts

Diagonal Matrices

Diagonal matrices are a special type of matrix in which all the elements outside of the main diagonal are zero. This means that the matrix has non-zero values only along its diagonal from the top left to the bottom right. Diagonal matrices are significant in mathematics because they are simple to work with and have straightforward properties.

Some properties of diagonal matrices include:

• The sum or product of two diagonal matrices is also a diagonal matrix.
• Multiplying a diagonal matrix by a scalar or another diagonal matrix results in another diagonal matrix.
• The eigenvalues of a diagonal matrix are the entries on its diagonal, making calculations involving them easy.

These characteristics make diagonal matrices convenient in various applications, such as solving linear equations or when dealing with transformations in linear algebra.

Matrix Algebra

Matrix algebra refers to a set of rules and operations that allow us to perform calculations involving matrices. These operations include addition, subtraction, multiplication, and finding inverses. Matrix algebra is fundamental in many applications ranging from physics to computer science as it enables efficient handling of large datasets and complex systems.

Key matrix operations are:

• Addition and subtraction: Matrices of the same dimensions can be added or subtracted by adding or subtracting corresponding elements.
• Multiplication: The product of two matrices is obtained by taking the dot product of rows and columns. This operation requires that the number of columns in the first matrix matches the number of rows in the second matrix.
• Transpose: Switching the rows and columns of a matrix.
• Inverse: A matrix that can "undo" the effect of a multiplication, provided that the original matrix is non-singular (i.e., its determinant is not zero).

Understanding these operations allows us to solve systems of equations, transform geometric data, and perform numerous other tasks in scientific computing.

Inverse of a Matrix

In matrix algebra, finding the inverse of a matrix is akin to finding a reciprocal for real numbers. If a matrix has an inverse, it is called "invertible" or "non-singular." For a square matrix \( A \), its inverse \( A^{-1} \) is a matrix such that their product is the identity matrix \( I \), i.e., \( AA^{-1} = A^{-1}A = I \).

Conditions and properties of matrix inverses include:

• A matrix must be square (same number of rows and columns) to have an inverse.
• Not all square matrices have inverses; those with zero determinants do not.
• The inverse of a diagonal matrix is easy to compute since it involves taking the reciprocal of each non-zero element on the diagonal while other elements remain zero.

When dealing with diagonal matrices, calculating the inverse is especially straightforward due to these properties. For example, the inverse of a diagonal matrix with elements \( a_1, a_2, ..., a_n \) is another diagonal matrix with elements \( a_1^{-1}, a_2^{-1}, ..., a_n^{-1} \), where each element is simply the reciprocal of the original element.