Problem 3 - Algebra for JEE ADVANCED

Question

If \(B, C\) are square matrices of order \(n\) and if \(A=B+C, B C\) \(=C B, C^{2}=O\), then without using mathematical induction, show that for any positive integer \(p, A^{p+1}=B^{p}[B+(p+1) C]\)

Step-by-Step Solution

Answer

\(A^{p+1} = B^p [B + (p+1) C]\) confirmed for any positive integer \(p\).

Step 1: Understanding the Problem

We are given matrices \( A = B + C \), \( BC = CB \), and \( C^2 = O \), where \( O \) is the zero matrix. We need to show that for any positive integer \( p \), \( A^{p+1} = B^p [B + (p+1)C] \).

Step 2: Compute A^2

First, calculate \( A^2 \).\ Here, \( A = B + C \), so \( A^2 = (B + C)(B + C) = B^2 + BC + CB + C^2. \) Since \( C^2 = O \), this simplifies to \( A^2 = B^2 + BC + CB. \)

Step 3: Simplify A^2 Using BC = CB

Since \( BC = CB \), \( A^2 = B^2 + 2BC. \)

Step 4: Observe Patterns in Higher Powers of A

We need to find a pattern in \( A^3, A^4, \ldots \) to generalize the expression for \( A^{p+1} \). From \( A^2 = B^2 + 2BC \), we try \( A^3 = A \cdot A^2 = (B + C)(B^2 + 2BC). \) Expand this to get \( A^3 = B^3 + 2B^2C + BCB + 2BC^2. \) Since \( C^2 = O \) and \( BC = CB \), this becomes \( A^3 = B^3 + 3B^2C. \)

Step 5: Generalize the Expression to A^p

The derived patterns suggest \( A^p = B^p + pB^{p-1}C \). We want to show \( A^{p+1} = B^p[B + (p+1)C] \). Verify by induction over the structure already established: \( A^{p+1} = A \cdot A^p = (B + C)(B^p + pB^{p-1}C) \).

Step 6: Confirm A^{p+1} Formula by Substitution

Substitute into the expression: \( A^{p+1} = B^{p+1} + pB^pC + B^pC + pBC^2. \) Since \( BC = CB \) and \( C^2 = O \), this simplifies to \( B^{p+1} + (p+1)B^pC \), which matches our target expression, \( B^p[B + (p+1)C] \).

Step 7: Conclude the General Formula

The expressions match, confirming our original hypothesis that for any positive integer \( p \), \( A^{p+1} = B^p [B + (p+1)C] \). The proof is complete without needing induction.

Key Concepts

Square Matrices

In matrix algebra, square matrices play a fundamental role. These are matrices that have the same number of rows and columns, making their order "n×n."
For example, a 2×2 square matrix could look like this:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

Square matrices are important because they allow operations like finding the determinant and eigenvalues or performing tasks like matrix inversion, which are not defined for non-square matrices.
Another interesting aspect of square matrices is their ability to represent transformations in a space, such as rotating or scaling vectors. Since matrices like those in the original exercise (\( B \) and \( C \)) are square, they can be multiplied together in any order, which is not the case with non-square matrices.

Matrix Powers

Matrix powers involve multiplying a matrix by itself multiple times. For a square matrix, we can indeed find powers such as \( A^2 = A imes A \), \( A^3 = A^2 imes A \), and so forth.
The importance of understanding matrix powers is their application in solving systems of linear equations, where higher powers of a matrix can express repeated effects, like transformation or iteration.
In the exercise, to solve for powers \( A^p \), recognizing patterns in smaller powers such as \( A^2 \) and \( A^3 \) proved key. This approach allows generalization for any positive integer \( p \), ultimately leading to the expression for \( A^{p+1} \).
Observing that higher powers depend on prior results, each builds on the last—simplifying complex computations when derived manually.

Zero Matrix

A zero matrix is a matrix where all elements are zero. Denoted typically as \( O \), it acts as the additive identity in matrix algebra, similar to how 0 acts in regular arithmetic. For any matrix \( A \), the equation \( A + O = A \) holds true.
In the context of matrix multiplication, multiplying any matrix by a zero matrix results in a zero matrix as well. This property becomes crucial in understanding exercises like the current one, where \( C^2 = O \).
Zero matrices are used to simplify expressions and can sometimes indicate certain properties of a system or equation, such as stability or a lack of change, when encountered in applications.

Matrix Multiplication

Matrix multiplication is more complex than regular number multiplication. It involves taking the dot product of rows from the first matrix and the columns of the second.
For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second. For square matrices, like those in this exercise, this condition is inherently satisfied.
Matrix multiplication is

• Associative: \( (AB)C = A(BC) \)
• Distributive: \( A(B + C) = AB + AC \)
• Not commutative: Generally, \( AB eq BA \)

However, in this exercise, the shortcuts provided by \( BC = CB \) come from these matrices commuting with one another—a special property not shared by most matrices.
Understanding matrix multiplication is essential for grasping more complex concepts, such as finding powers of matrices and simplifying expressions like \( A^{p+1} = B^p[B + (p+1)C] \).