# Is B Equal to A³ Given Symmetric and Invertible Matrices?

• MHB
• Yankel
In summary, the conversation discusses solving a mathematical problem involving squared invertible matrices and symmetric matrices. The correct approach is to multiply both sides by different matrices in each iteration. It is also mentioned that with a symmetric and invertible matrix, the inverse is not necessarily equal to the original matrix.
Yankel
Hello all,

If A and B are both squared invertible matrices and A is also symmetric and:

$AB^{-1}AA^{T}=I$

Can I say that

$B=A^{3}$ ?

In every iteration of the solution, I have multiplied both sides by a different matrix. At first by the inverse of A, then the inverse of the transpose, etc...Is this the correct approach to solve this ? Thank you in advance !

Another question. If A in both symmetric and invertible, it doesn't mean that the inverse of A is equal to A, right ?

Yankel said:
Hello all,

If A and B are both squared invertible matrices and A is also symmetric and:

$AB^{-1}AA^{T}=I$

Can I say that

$B=A^{3}$ ?

In every iteration of the solution, I have multiplied both sides by a different matrix. At first by the inverse of A, then the inverse of the transpose, etc...Is this the correct approach to solve this ? Thank you in advance !

Sure.
With $A$ symmetric and invertible, we have indeed:
$$AB^{-1}AA^{T}=I \quad\Rightarrow\quad AB^{-1}AA=I \quad\Rightarrow\quad A^{-1}AB^{-1}AA=A^{-1}I \quad\Rightarrow\quad B^{-1}AA=A^{-1} \\ \quad\Rightarrow\quad BB^{-1}AAA=BA^{-1}A \quad\Rightarrow\quad AAA=B \quad\Rightarrow\quad B=A^3$$

Yankel said:
Another question. If A in both symmetric and invertible, it doesn't mean that the inverse of A is equal to A, right ?

Nope.
That will only be the case if $A$ is either identity or a reflection (only eigenvalues $\pm 1$).

Thanks !

## 1. What is a symmetric matrix?

A symmetric matrix is a square matrix in which the elements are symmetric about the main diagonal. This means that the elements at position (i,j) and (j,i) are equal. In other words, a symmetric matrix is equal to its transpose.

## 2. What are the properties of a symmetric matrix?

Some properties of a symmetric matrix include:

• All eigenvalues of a symmetric matrix are real.
• A symmetric matrix is always diagonalizable.
• The sum of two symmetric matrices is also a symmetric matrix.
• The product of two symmetric matrices is a symmetric matrix only if the matrices commute.

## 3. How can I determine if a matrix is symmetric?

To determine if a matrix is symmetric, you can check if it is equal to its transpose. This can be done by comparing the elements at position (i,j) and (j,i) for all values of i and j. If they are equal, the matrix is symmetric.

## 4. What is an invertible matrix?

An invertible matrix, also known as a non-singular matrix, is a square matrix that has an inverse. This means that there exists another matrix that, when multiplied with the original matrix, results in the identity matrix. In other words, an invertible matrix is one that can be "undone".

## 5. How can I determine if a matrix is invertible?

A matrix is invertible if its determinant is non-zero. This can be checked by calculating the determinant of the matrix using various methods, such as using cofactors or row reduction. If the determinant is non-zero, the matrix is invertible. Additionally, a square matrix is invertible if its columns are linearly independent.

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