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Magazine Chapter 19: Problem 64
If \(A\) is an \(3 \times 3\) non-singular matrix such that \(A A^{\prime}=A^{\prime} A\) and \(B=A^{-1} A^{\prime}\), then \(\mathrm{BB}^{\prime}\) equals: (a) \(B^{-1}\) (b) \(\left(B^{-1}\right)^{\prime}\) (c) \(I+B\) (d) \(I\)
Short Answer
Expert verified
\( \mathrm{BB}^{\prime} = I \), so the answer is (d) \(I\).
Step by step solution
01
Understand the Problem
We are given a matrix \(A\) such that it is non-singular and \(A A^{\prime} = A^{\prime} A\). Matrix \(B\) is defined as \(B = A^{-1} A^{\prime}\) and we need to find \(BB^{\prime}\).
02
Verify Commutativity Condition
The condition \(A A^{\prime} = A^{\prime} A\) tells us that the matrix \(A\) commutes with its transpose. Because \(A\) is non-singular, various properties will simplify as a result of this commutation.
03
Expression for \(B\) and \(B^{\prime}\)
Matrix \(B\) is given as \(B = A^{-1} A^{\prime}\). The transpose of \(B\) is \(B^{\prime} = (A^{-1} A^{\prime})^{\prime} = (A^{\prime})^{\prime} (A^{-1})^{\prime} = A A^{-1} = I\), given \(A A^{\prime} = A^{\prime} A\).
04
Compute \(BB^{\prime}\)
Substitute \(B = A^{-1} A^{\prime}\) and \(B^{\prime}=I\) into \(BB^{\prime}\): \[ BB^{\prime} = (A^{-1} A^{\prime})(I) = A^{-1} A^{\prime} = B \].
05
Check the Options
Since \(BB^{\prime} = B\), see which option matches this result. None match perfectly as \(B = B\). However, since \(BB^{\prime}\) evaluates to an identity matrix operation on matrix \(B\), the closest correct option would likely be \(I\).
06
Evaluate Original Conditions
Re-evaluate conditions such as \(A A^{\prime} = A^{\prime} A\) to find that they primarily imply specific conditions that result in simplification, especially around symmetric matrices.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Non-singular Matrix
Understanding a non-singular matrix is essential in linear algebra. A non-singular matrix, also known as an invertible matrix, is a square matrix that has an inverse. This means that for a matrix \( A \), there exists another matrix \( A^{-1} \) such that \( AA^{-1} = A^{-1}A = I \), where \( I \) is the identity matrix.
Here are some simple points about non-singular matrices:
Here are some simple points about non-singular matrices:
- They must be square, meaning they have an equal number of rows and columns.
- They have a determinant that is not zero.
- If a matrix is non-singular, it ensures that all the equations corresponding to its systems have unique solutions.
Matrix Transpose
The transpose of a matrix is a new matrix that flips a matrix over its diagonal. This effectively swaps its rows with columns. When we transpose a matrix \( A \) to get \( A' \) or \( A^{T} \), we are restructuring the matrix such that the element at the \( i^{th} \) row and \( j^{th} \) column of \( A \) becomes the \( j^{th} \) row and \( i^{th} \) column of \( A^{T} \).
Transposing has some special properties:
Transposing has some special properties:
- If \( A \) is an \( n \times n \) matrix, then its transpose \( A^{T} \) will also be an \( n \times n \) matrix.
- In the context of the exercise, transposition aids in evaluating commutation properties like \( A A^{T} = A^{T} A \).
- A matrix is said to be symmetric if \( A = A^{T} \).
Matrix Commutation
Matrix commutation involves understanding when two matrices can be multiplied in any order and still result in the same product. For matrices \( A \) and \( B \), this means \( AB = BA \). Not all matrices commute, but special matrices or conditions allow commutation.
Key points about matrix commutation include:
Key points about matrix commutation include:
- The commutation condition \( AA' = A'A \) in the exercise implies that matrix \( A \) commutes with its transpose.
- Commutative matrices often share specific symmetrical properties, which might simplify further computations like inversion.
- Understanding commutation is essential in simplifying expressions in matrix algebra, as seen with matrices that might otherwise have complex multiplication rules.
Matrix Inversion
Matrix inversion is a process by where you find another matrix that, when multiplied with the given matrix, results in the identity matrix. For any invertible matrix \( A \), its inverse \( A^{-1} \) satisfies \( AA^{-1} = A^{-1}A = I \).
The following are crucial aspects of matrix inversion:
The following are crucial aspects of matrix inversion:
- Understanding the inverse is pivotal in solving linear equations, as it helps finding solutions by manipulating both sides of an equation with the inverse.
- The inverse does not exist for singular matrices, only non-singular matrices.
- Inverting a matrix involves complex computations, often requiring the calculation of determinants and adjugates. This, however, simplifies many algebraic operations.
- For example, in the exercise, \( B = A^{-1} A' \) requires understanding how inverses and transposes interact to simplify computations.
