Problem 253 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(B\) is idempotent, show that \(A=I-B\) is also idempotent and that \(A B=B A=O\).

Step-by-Step Solution

Answer

As per our analysis and step-by-step solution, it is clear that the matrix \(A\) is indeed idempotent because \(A^2 = A\) and that \(AB = 0\).

Step 1: Define Idempotent

An idempotent matrix is one that, when multiplied by itself, gives a result equal to itself, i.e. if matrix \(B\) is idempotent, \(B^2=B\). The first task is to show \(A^2=A\). Given \(A = I - B\) where \(I\) is the identity matrix, substitute in the definition of A.

Step 2: Calculate \(A^2\) and simplify the equation

\To calculate \(A^2\), multiply \(A\) by itself. So, \(A^2 = (I - B)(I - B) = I^2 - 2IB + B^2 = I - 2B + B\), since \(B^2\) equals \(B\) for an idempotent matrix and \(I^2\) equals \(I\) for the Identity matrix. Moreover, simplifying the equation yields \(A^2 = I - B = A\), proving that \(A\) is idempotent.

Step 3: Verify that AB = B

Given, \(A = I - B\), calculate the multiplication \(AB\) by replacing \(A\) in the equation: \(AB = (I - B)B = IB - B^2 = B - B = 0\), since identity matrix \(I\) times any matrix equals the same matrix, and and \(B^2 = B\) for an idempotent matrix \(B\)

Key Concepts

Matrix Multiplication

Matrix multiplication is foundational in linear algebra and is used to combine two matrices, resulting in a new matrix. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The product of a matrix A of size m x n and a matrix B of size n x p is a matrix C of size m x p. The entry c_ij in the product matrix is calculated by taking the dot product of the ith row of A and the jth column of B.

As an example, consider the matrix A and its multiplication with B in our exercise:

• If B is idempotent, by definition, B² (the matrix B multiplied by itself) equals B. When multiplying A (which is I - B) by B, we follow the rules of matrix multiplication to find that AB = (I - B)B. After distributing and simplifying, we get the zero matrix, symbolized as O, showcasing that in this special case, the product is a matrix of zeroes.

This process demonstrates the elegance and complexity of matrix multiplication in linear algebra.

Identity Matrix

The identity matrix, usually denoted as I, is crucial in matrix operations. It is defined as a square matrix with ones on the diagonal and zeros everywhere else. Its role is similar to the number 1 in scalar multiplication; when any matrix A is multiplied by the identity matrix, the result is A itself. This property is key in many proofs and algebraic manipulations.

In our example, the identity matrix plays an integral part as A is defined as I - B. When calculating A², we use the fact that I² = I to simplify our expression. The identity matrix's simplicity provides a foundation that can be used to explore more complex matrix behavior, such as the idempotent matrices discussed in the exercise.

Linear Algebra

Linear algebra is a field of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It is fundamentally about understanding and manipulating spaces defined by linear relationships. Idempotent matrices are one of many special types of matrices studied in this field. As shown in our exercise, proving that a matrix is idempotent involves matrix manipulation using the principles of linear algebra.

Understanding the properties and implications of operations like multiplying by the identity matrix or confirming the idempotent property are vital skills in linear algebra. These concepts not only help solve theoretical problems but also have practical applications in areas such as computer graphics, machine learning, and quantum mechanics. Linear algebra provides the language and tools for many domains where systems are modeled and solved mathematically.