Problem 20 - Algebra for JEE ADVANCED

Question

If \(A\) is a non-singular matrix such that \(A A^{T}=A^{T} A\) and \(B=A^{-1} A^{\tau}\), then matrix \(B\) is a. involuntary b. orthogonal c. idempotent d. none of these

Step-by-Step Solution

Answer

Matrix B is orthogonal.

Step 1: Understanding Matrix Properties

We need to use the given property of matrix \( A \), specifically that \( AA^{T}=A^{T}A \). Since \( A \) is non-singular, it is invertible, allowing us to define the matrix \( B = A^{-1} A^{\tau} \). We need to check the properties of \( B \).

Step 2: Express Transpose and Inverse

Express \( A^{\tau} \) which is the same as \( A^{T} \), meaning \( B = A^{-1}A^{T} \).

Step 3: Check if B is Idempotent

To be idempotent, \( B imes B = B \). Calculate \( B^2 = (A^{-1}A^{T})(A^{-1}A^{T}) = A^{-1}(A^{T}A)A^{T} \). Since \( A^{T}A = AA^{T} \), this simplifies to \( A^{-1}AA^{T}A^{T} = A^{T}A^{T} \), and simplifying further does not return \( B \). Thus \( B \) is not idempotent.

Step 4: Check if B is Orthogonal

For \( B \) to be orthogonal, \( B B^{T} = I \). Calculate \( B B^{T} = (A^{-1}A^{T})(A^{T{-1}}(A^{-1})^{T}) \). Since \((A^{-1})^{T} = (A^{T})^{-1}\), this reduces to \( (A^{-1}A^{T}A^{-1})(A^{T})^{-1} = A^{-1}(AA^{-1})A^{-1}(A^{T})^{-1} = I \). Hence, \( B \) is orthogonal.

Key Concepts

Non-Singular Matrix

A non-singular matrix, also known as an invertible or nonsingular matrix, is pivotal in linear algebra. It is a square matrix that has an inverse. This means for a matrix \( A \), there exists another matrix \( A^{-1} \) such that \[ AA^{-1} = A^{-1}A = I \]where \( I \) is the identity matrix. This property is crucial for solving systems of linear equations and finding the inverse of transformation operations.

Non-singular matrices have several key properties:

• The determinant of a non-singular matrix is always non-zero.
• They allow for unique solutions when used in systems of equations.
• Operations such as matrix multiplication involving non-singular matrices and their inverses adhere strictly to the associative and distributive laws.

Understanding these properties helps in determining characteristics and behaviors of other matrices, such as orthogonal or idempotent matrices.

Orthogonal Matrix

An orthogonal matrix is a fascinating type of square matrix whose rows and columns are orthonormal vectors. This means its inverse is simply its transpose, and it preserves the dot products. For a matrix \( B \) to be orthogonal, it must satisfy the condition:\[ B B^{T} = B^{T} B = I \]where \( I \) is the identity matrix.

Orthogonal matrices are essential in various applications:

• They maintain the length and angles of vectors they transform, making them critical in geometry and computer graphics.
• Calculations with orthogonal matrices can be highly informative as they simplify many matrix operations such as reductions and decompositions.
• They are stable in numeric computations, reducing errors in iterative methods.

By exploring the orthogonal property in matrices, one gains insights into stable computations and the elegant symmetries of linear transformations.

Idempotent Matrix

An idempotent matrix is a matrix that, when multiplied by itself, yields the same matrix. In mathematical terms, a matrix \( B \) is idempotent if:\[ B^2 = B \]This property is particularly intriguing because it links to projection operations and eigenvalues in linear algebra.

Properties of idempotent matrices include:

• The eigenvalues of an idempotent matrix are either 0 or 1, which provides significant information about the matrix's rank and nullity.
• Idempotent matrices are crucial in statistical applications, specifically in regression analysis as they represent projection operators.
• They are utilized in optimization problems where maintaining or reducing dimensions without altering essential properties is needed.

By understanding idempotent matrices, one can delve into sophisticated concepts of linear transformations and their roles in various mathematical and real-world applications.