Problem 76 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\mathrm{A}\) is a \(3 \times 3\) non-singular matrix such that \(A A^{\prime}=\) \(A^{\prime} A\) and \(B=A^{-1} A^{\prime}\), then \(B B^{\prime}\) equals \(\quad\) [2014] (A) \(\mathrm{I}+\mathrm{B}\) (B) I (C) \(B^{-1}\) (D) \(\left(B^{-1}\right)^{\prime}\)

Step-by-Step Solution

Answer

Option (B): I.

Step 1: Identifying Given Conditions

We are given that matrix \( A \) is a \( 3 \times 3 \) non-singular matrix. It satisfies the condition \( AA' = A'A \), meaning \( A \) commutes with its transpose. Additionally, \( B = A^{-1}A' \). Our task is to find \( BB' \).

Step 2: Using the Properties of A

Since \( A \) is a \( 3 \times 3 \) non-singular matrix and \( AA' = A'A \), we can conclude that \( A \) could be a normal matrix (as symmetric matrices satisfy \( AA' = A'A \), but normal matrices satisfy the commutation property universally).

Step 3: Calculating B's Transpose

Calculate \( B' \) which is the transpose of \( B = A^{-1}A' \). Thus, \( B' = (A^{-1}A')' = (A')' (A^{-1})' = A (A^{-1})' \), due to the property \((XY)' = Y'X'\). Since \((A^{-1})' = (A')^{-1}\), we have \( B' = A A^{-1'} \).

Step 4: Simplifying B'

Since \( A \) is a normal matrix condition \((AA') = (A'A)\), \( A^{-1'} = (A')^{-1} \) thereby simplifying \( B' \) to \( B' = AA^{-1} = I \) where \( I \) is the identity matrix.

Step 5: Calculating BB'

We now calculate \( BB' = (A^{-1}A')(A^{-1}A')' = A^{-1}A'AA^{-1}A = A^{-1}IA^{-1} = A^{-1}A^{-1} = I \), considering \( (A')(A^{-1}) = I \). But with this matching identity transformations we'll have \( BB' = I \).

Step 6: Confirming Result

According to our calculation, \( BB' = I \), which satisfies one of the multiple choice options directly. Thus, the response is (B) I.

Key Concepts

Non-singular Matrix

A non-singular matrix, also known as an invertible matrix, is a matrix that has a unique matrix inverse. This means that the matrix can be expressed as a full-rank matrix, indicating that its determinant is non-zero. When dealing with matrices in mathematics, a non-singular matrix is primarily distinguished by this property.

Some of the key features of a non-singular matrix include:

• Existence of an inverse, denoted by the same matrix symbol with a superscript "-1" (e.g., if the matrix is known as \( A \), the inverse is \( A^{-1} \)).
• An important characteristic is that multiplication of a non-singular matrix by its inverse yields the identity matrix \(I\).
• For square matrices, being non-singular is equivalent to having a non-zero determinant.
• All non-singular matrices are a part of the set of non-degenerate matrices.

Understanding non-singular matrices is crucial for solving linear equations and analyzing mathematical systems.

Matrix Transpose

The transpose of a matrix is a new matrix obtained by switching rows with columns. This operation is denoted by a prime symbol, e.g., the transpose of matrix \( A \) is represented as \( A' \). It's an essential concept in matrix algebra, often used in calculations involving symmetry and optimization.

Significant properties of matrix transpose include:

• The transpose of a transpose matrix returns the original matrix: \((A')' = A \).
• Transposing a product of matrices reverses their order: \((AB)' = B'A'\).
• Transpose of a matrix sum conforms to: \((A + B)' = A' + B'\).
• If a matrix is square and symmetric, then \(A = A'\).

The transpose operation helps analyze matrix alignments and plays a vital role in concepts like orthogonal matrices and determinants.

Matrix Inverse

The inverse of a matrix is a concept wherein a given matrix \(A\), its inverse \(A^{-1}\), when multiplied together, yield the identity matrix: \(AA^{-1} = A^{-1}A = I\). This property resembles the way numbers function in arithmetic, where a number times its reciprocal equals one.

Key points about matrix inverses:

• Not all matrices have inverses; a matrix must be non-singular (determinant ≠ 0) to have an inverse.
• The inverse operation reverses the effect of a matrix multiplication, which is important for solving linear equations like \(AX = B\).
• The inverse of a product of matrices can be obtained by reversing the order of their inverses: \((AB)^{-1} = B^{-1}A^{-1}\).
• The concept extends to solving systems of equations where \(X = A^{-1}B\).

Matrix inverse is a vital tool in computations involving transformations and systems modeling, making it a cornerstone of linear algebra.

Identity Matrix

An identity matrix is a pivotal element in matrix algebra, known for its unique property of not altering any matrix it's multiplied by. It's a square matrix with ones on the main diagonal and zeroes elsewhere, denoted as \(I\). This matrix acts similarly to the number "1" in multiplication.

Some core characteristics include:

• Any matrix \(A\) multiplied by the identity matrix returns the original matrix: \(AI = IA = A\).
• Identity matrices are used in defining matrix inverses, as they help revert a matrix to its original form in multiplication.
• They serve as the neutral element in matrix multiplication and are part of defining unitary and orthogonal matrices.
• The identity matrix \(I\) conforms to different sizes, each adapting to the size of \(A\) it's being multiplied with (e.g., a 3x3 identity matrix for a 3x3 matrix).

Understanding identity matrices is fundamental for manipulating and simplifying matrix equations, significantly aiding in subjects like computer graphics and engineering.