Problem 4 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A\) is an involutory matrix and \(I\) is unit matrix of the same order then, \((I-A)(I+A)=\) (A) 0 (B) \(A\) (C) \(I\) (D) \(2 A\)

Step-by-Step Solution

Answer

(I-A)(I+A) = 0, so the answer is (A) 0.

Step 1: Identify Involutory Matrix

An involutory matrix is defined as a matrix that is its own inverse. Thus, if \( A \) is an involutory matrix, then \( A^2 = I \) where \( I \) is the identity matrix of the same order as \( A \).

Step 2: Set up the Expression to Simplify

The problem requires us to simplify \((I-A)(I+A)\). We recognize this as a difference of squares type expression, which can be simplified using the identity \((x-y)(x+y) = x^2 - y^2\).

Step 3: Apply Difference of Squares

Given the expression \((I-A)(I+A) = I^2 - A^2\) as in the identity \( x^2 - y^2 \).

Step 4: Substitute Using Involutory Property

Since \(A\) is involutory, \(A^2 = I\). Substitute \(A^2 = I\) into the simplified expression, giving us \(I^2 - I = I - I = 0\).

Step 5: Choose the Correct Option

Comparing the result to the provided options, we see the correct answer is \( (I-A)(I+A) = 0 \). This corresponds to option (A) 0.

Key Concepts

Identity Matrix

The identity matrix, often denoted as \( I \), is a crucial concept in understanding matrices. It acts like the number 1 in matrix operations. Here's why it's so important:

• For any matrix \( A \), multiplying it by the identity matrix \( I \) of the same order (either on the left or the right) will return \( A \) itself, fulfilling the equation \( AI = IA = A \).
• An identity matrix is a square matrix, meaning it has the same number of rows and columns.
• All the diagonal entries of an identity matrix are 1, while all other entries are 0. This unique structure is what allows it to act as a multiplicative identity element in matrix arithmetic.

Understanding the identity matrix is key to grasping more complex topics like involutory matrices or matrix inversion. For example, in our problem, the property \( A^2 = I \) shows the role of the identity matrix in defining involutory matrices.

Matrix Inverse

When we talk about the matrix inverse, we're referring to a matrix \( A^{-1} \) which, when multiplied with the original matrix \( A \), results in the identity matrix \( I \). This can be written as:

• \( AA^{-1} = A^{-1}A = I \)

The concept of a matrix inverse is somewhat similar to dividing numbers. However, not all matrices have inverses. A matrix must be square and non-singular (having a non-zero determinant) to be invertible. Involutory matrices serve as a special case. They are their own inverses, meaning if \( A \) is involutory, then \( A = A^{-1} \). This property is rooted in the equation \( A^2 = I \), asserting that multiplying \( A \) by itself yields the identity matrix. Having this property simplifies calculations greatly, as shown in our exercise, where manipulations with an involutory matrix lead to simple outcomes.

Difference of Squares

The difference of squares is a fundamental algebraic identity that can be applied to matrices in a straightforward way. The identity states:

• \((x-y)(x+y) = x^2 - y^2\)

This formula helps simplify expressions that follow this pattern, regardless of whether \( x \) and \( y \) are numbers or matrices.In the given exercise problem, the expression \((I-A)(I+A)\) follows this exact pattern. By recognizing this, we simplify it to \( I^2 - A^2 \). Because \( A \) is an involutory matrix satisfying \( A^2 = I \), substituting back gives \( I^2 - I = 0 \).This demonstrates how the difference of squares identity can make solving matrix problems faster and cleaner, particularly when combined with properties of special matrices like the involutory matrix.