Problem 51 - Algebra for JEE ADVANCED

Question

If \(A\) and \(B\) are square matrices of the same order and \(A\) is nonsingular, then for a positive integer \(n,\left(A^{-1} B A\right)^{\prime}\) is equal to \(\begin{array}{ll}\text { a. } A^{-n} B^{n} A^{n} & \text { b. } A^{\prime \prime} B^{\prime \prime} A^{-\pi}\end{array}\) c. \(A^{-1} B^{n} A\) d. \(n\left(A^{-1} B A\right)\)

Step-by-Step Solution

Answer

The expression for \((A^{-1} B A)^n\) does not match any options. Recalculate it.

Step 1: Understanding the Problem

We need to find the transpose of \((A^{-1} B A)^n\) given \(A\) is a nonsingular matrix (meaning it has an inverse). The transpose of a product of matrices has specific properties which we'll utilize.

Step 2: Recall Properties of Transpose

Recall that the transpose of a product \( (XY)^T \) is equal to \( Y^T X^T \). For multiple products, \((A B C)^T = C^T B^T A^T\). We'll use this property extensively.

Step 3: Obtain Transpose of Single Product

Given the expression \( (A^{-1} B A)^n \), first consider \( A^{-1} B A \). The transpose of this product using the properties is \( A^T (B^T (A^{-1})^T) \).

Key Concepts

Transpose of Matrices

A matrix transpose is a crucial concept in linear algebra that involves flipping a matrix over its diagonal. In simple terms, the rows of the original matrix become columns, and vice versa. This is especially useful when working with operations that require adjustments in orientation.
For any given matrix \( A \), the transpose is denoted as \( A^T \). A handy property to remember is that the transpose of a product of matrices is the reverse of the product of their transposes. For example, for matrices \( X \) and \( Y \), the transpose \( (XY)^T \) is equal to \( Y^T X^T \).

• This can be extended to multiple matrices, such that for \( A \), \( B \), and \( C \), the relation \( (ABC)^T = C^T B^T A^T \) holds true.


• Another key property is that the transpose of a transpose brings you back to the original matrix: \( (A^T)^T = A \).

When solving problems involving matrix transposition, like the given problem, it’s essential to apply these properties correctly.

Properties of Inverse Matrices

Matrix inversion is another fundamental concept in matrix algebra. An inverse matrix is akin to the reciprocal of a number, aiming to return the product to an identity when multiplied by the original matrix. Consider a square matrix \( A \); if there exists another matrix \( A^{-1} \) such that \( AA^{-1} = A^{-1}A = I \), where \( I \) is the identity matrix, then \( A^{-1} \) is called the inverse of \( A \).
The inverse of a matrix has several practical properties:

• Only square matrices (having equal number of rows and columns) can have an inverse.


• The inverse of a product of two matrices is the product of their inverses in the reverse order: \( (AB)^{-1} = B^{-1} A^{-1} \).


• The inverse of a matrix exists only if the determinant of the matrix is non-zero; such matrices are termed as nonsingular.

Understanding these properties is vital when working with equations that involve inverses, ensuring you can manipulate expressions effectively, as seen in the problem with \( (A^{-1} B A) \).

Matrix Exponentiation

Matrix exponentiation involves raising a matrix to a power, much like numerical exponentiation. For a square matrix \( A \), raising it to a positive integer power \( n \) means multiplying \( A \) by itself \( n \) times: \( A^n = A \times A \times \ldots \times A \) (\( n \) times).
Some pivotal properties to keep in mind:

• The operation is primarily defined for square matrices.


• If \( A \) is an invertible (nonsingular) matrix, \( (A^{-1})^n \) will be equivalent to \( (A^n)^{-1} \).


• Matrix exponentiation is associative, meaning \( (A^m)^n = A^{m*n} \).

In the context of the exercise, understanding and applying matrix exponentiation correctly can simplify complex matrix products, particularly when combined with matrix transposition and inversion properties. When faced with tasks like calculating \( \left( A^{-1} B A \right)^n \), combining knowledge of these three core concepts ensures you're well-prepared to tackle such challenges.