Math Problem Statement
If A \in A_{F} is invertible, prove that the transpose of A ^ - 1 is the inverse of the transpose of A.
Solution
To prove that the transpose of is the inverse of the transpose of , we proceed as follows:
Theorem:
If is an invertible matrix, then:
Proof:
Step 1: Property of Invertible Matrices
Since is invertible, we know that:
where is the identity matrix.
Step 2: Taking the Transpose
Take the transpose of both sides of the equation :
Since the transpose of the identity matrix is itself (), we get:
Step 3: Using the Property of Transpose of a Product
The transpose of a product of matrices satisfies:
Applying this to , we have:
Thus, the equation becomes:
Step 4: Definition of the Inverse
By definition, a matrix is the inverse of if:
From the equation , it follows that:
Conclusion:
We have proved that:
Thus, the transpose of is indeed the inverse of the transpose of .
Would you like further clarification or examples related to this proof?
5 Related Questions:
- What is the transpose of a product of two matrices, and how is it derived?
- How do you prove that the transpose of the identity matrix is itself?
- Can we generalize this property to block matrices or higher dimensions?
- How does the determinant of relate to the invertibility of ?
- How can we verify this theorem numerically for a 2x2 matrix?
Tip:
When working with matrix inverses, always check if the determinant is non-zero to ensure the matrix is invertible.
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Transpose of a Matrix
Matrix Inverse
Formulas
(A B)^T = B^T A^T
(A^{-1})^T = (A^T)^{-1}
Theorems
Property of Matrix Transpose
Definition of Matrix Inverse
Suitable Grade Level
Undergraduate Mathematics or Advanced High School (Grades 11-12)
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