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 A1A^{-1} is the inverse of the transpose of AA, we proceed as follows:


Theorem:

If AAFA \in A_F is an invertible matrix, then:

(A1)T=(AT)1.(A^{-1})^T = (A^T)^{-1}.


Proof:

Step 1: Property of Invertible Matrices

Since AA is invertible, we know that:

AA1=I,A A^{-1} = I,

where II is the identity matrix.

Step 2: Taking the Transpose

Take the transpose of both sides of the equation AA1=IA A^{-1} = I:

(AA1)T=IT.(A A^{-1})^T = I^T.

Since the transpose of the identity matrix II is itself (IT=II^T = I), we get:

(AA1)T=I.(A A^{-1})^T = I.

Step 3: Using the Property of Transpose of a Product

The transpose of a product of matrices satisfies:

(AB)T=BTAT.(A B)^T = B^T A^T.

Applying this to (AA1)T(A A^{-1})^T, we have:

(AA1)T=(A1)TAT.(A A^{-1})^T = (A^{-1})^T A^T.

Thus, the equation becomes:

(A1)TAT=I.(A^{-1})^T A^T = I.

Step 4: Definition of the Inverse

By definition, a matrix BB is the inverse of ATA^T if:

BAT=I.B A^T = I.

From the equation (A1)TAT=I(A^{-1})^T A^T = I, it follows that:

(A1)T=(AT)1.(A^{-1})^T = (A^T)^{-1}.


Conclusion:

We have proved that:

(A1)T=(AT)1.(A^{-1})^T = (A^T)^{-1}.

Thus, the transpose of A1A^{-1} is indeed the inverse of the transpose of AA.


Would you like further clarification or examples related to this proof?


5 Related Questions:

  1. What is the transpose of a product of two matrices, and how is it derived?
  2. How do you prove that the transpose of the identity matrix is itself?
  3. Can we generalize this property to block matrices or higher dimensions?
  4. How does the determinant of AA relate to the invertibility of ATA^T?
  5. 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)