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Magazine Chapter 5: Problem 76
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) \(\left(B^{-1}\right)^{\prime}\)
Short Answer
Expert verified
Option (B): I.
Step by step solution
01
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 \( BB' \).
02
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).
03
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 \( B' = A A^{-1'} \).
04
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.
05
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 \( BB' = I \).
06
Confirming Result
According to our calculation, \( BB' = I \), which satisfies one of the multiple choice options directly. Thus, the response is (B) I.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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 \(A = A'\).
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:
Key points about matrix inverses:
- Not all matrices have inverses; a matrix must be non-singular (determinant 鈮� 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 \(X = A^{-1}B\).
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:
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).
