Problem 28 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Matrix \(A\) is such that \(A^{2}=2 A-I\), where \(I\) is unit matrix then for \(n \geq 2, A^{n}=\) (A) \(n A-(n-1) I\) (B) \(n A-I\) (C) \(2^{n-1} A-(n-1) I\) (D) \(2^{n-1} A-I\)

Step-by-Step Solution

Answer

The correct expression is (A) \(n A - (n-1) I\).

Step 1: Understanding the given equation

The equation given is \(A^2 = 2A - I\), where \(A\) is a matrix and \(I\) is the identity matrix. We need to use this equation to express \(A^n\), for \(n \geq 2\).

Step 2: Rewriting the equation

From the equation \(A^2 = 2A - I\), we can rearrange it into a recurrence form: \(A^2 = 2A - I\). We aim to derive higher powers of \(A\) based on this relation.

Step 3: Calculating further powers of A using induction

Assume that \(A^k = kA - (k-1)I\) holds true for some \(k\), and we will prove it for \(k + 1\). Then \(A^{k+1} = A \cdot A^k = A(kA - (k-1)I) = kA^2 - (k-1)A.\)

Step 4: Substituting the given equation into the induction step

Using \(A^2 = 2A - I\) in \(A^{k+1} = kA^2 - (k-1)A\), we have \(A^{k+1} = k(2A - I) - (k-1)A = 2kA - kI - (k-1)A.\)

Step 5: Simplifying the expression

Simplify \(A^{k+1} = (2k - (k-1))A - kI\), which becomes \((k + 1)A - kI\). This confirms \(A^{k+1} = (k+1)A - kI\) holds true.

Step 6: Concluding with the derived formula

Since the induction holds, the formula for \(A^n\) is \(A^n = nA - (n-1)I\) for \(n \geq 2\).

Key Concepts

Recurrence Relation

A recurrence relation expresses each term of a sequence as a function of its preceding terms. This is a crucial concept in sequences involving matrix exponentiation, where the power of a matrix can be expressed in relation to its previous powers.

In our problem, we have the equation:

• Given: \( A^2 = 2A - I \).

This equation is the foundation to establish a recurrence relation for powers of matrix \( A \). Essentially, each power—\( A^n \)—relies on the form derived from \( A^2 \).

To derive the higher powers of \( A \), we continue to apply this pattern, ensuring every new expression includes the previous powers, precisely as in a sequence governed by recurrence relations.

Inductive Proof

Inductive proof is a method used to prove statements about all natural numbers. To prove a statement by induction, you typically follow these steps:

• **Base Case:** Verify the statement holds for the initial value, often \( n=2 \) for problems involving sequences starting from this point.
• **Inductive Step:** Assume it holds for some natural number \( k \), and then demonstrate it holds for \( k+1 \).

For our problem, we begin by verifying for \( n=2 \):

• We know \( A^2 = 2A - I \), aligning with our proposed form, confirming the base case.

Next, the inductive step ensures the expression \( A^k = kA - (k-1)I \) leads to the valid conclusion of \( A^{k+1} = (k+1)A - kI \). Through this method, every step is proven based on its predecessor, ensuring the formula holds for all \( n \geq 2 \).

This logic of induction is integral in proving patterns like the one derived in the exercise.

Identity Matrix

The identity matrix, denoted by \( I \), is like the number 1 in matrix algebra. It possesses unique properties that simplify matrix equations. It is a square matrix with ones on the diagonal and zeros elsewhere.

• **Property:** \( A \cdot I = A \).
• **Relevance:** The equation \( A^2 = 2A - I \) entails subtracting this identity matrix, which impacts the expressions for subsequent powers of \( A \).

The identity matrix ensures that operations begin or maintain the original matrix's structural integrity. In our specific solution, using \( I \) in the relation shows how \( I \)'s role affects the balance and formulation of \( A^n \). By understanding this, solving problems involving identities becomes more intuitive, as they help maintain the form and result.

Matrix Algebra

Matrix algebra is the foundation of numerous operations and transformations in mathematics, providing a framework for manipulating matrices through addition, multiplication, scalar products, and more.

• **Matrix Multiplication:** Matrices can be multiplied, following specific rules, and this operation is non-commutative.
• **Application:** The equation \( A^2 = 2A - I \) and calculating \( A^{k+1} \) are done using these principles.

Understanding matrix algebra is crucial because it provides the rules that govern how to handle matrices like \( A \) in complex expressions. For instance, multiplying a matrix by the identity matrix, or adding matrices, follows defined rules ensuring clarity and simplicity in deriving higher expressions for \( A^n \). Each calculation respects the properties of matrices, ensuring logical outcomes when solving algebraic expressions of this nature.