Matrices Inverse properties

In summary, we are trying to determine which of the given formulas hold true for all invertible nxn matrices A and B. We know that 2A is always invertible and that (A+B)^2 is not always equal to A^2 + B^2 + 2AB. To prove this, we can consider looking at (A+A^-1)^7 and expanding it. For formula 5, we can think of examples where A and B are both invertible, but their sum is not. And for formula 6, we need to consider the self-similarity of B with respect to A in order for ABA^-1 to be equal to B.
  • #1
thepassenger48
13
0

Homework Statement



Determine which of the formulas hold for all invertible nxn matrices A and B

1. (A+A^-1)^7 = A^7+A^-7
2. 2A is invertible
3. (A+B)² = A² + B² + 2AB
4. (ABA^-1)^6 = AB^6A^-1
5. A + B is invertible
6. ABA^-1 = B


I know 2A should be invertible, and number 3 wrong, but what else?
Thanks
 
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  • #2
Try to think of examples if a fact bothers you or if you can't proove it.
 
  • #3
For 1) and 3, write out the computation- for 1 you might want to look at (A+ A-1)2 first.

For 5, if A is invertible, then so is B= -A.
 
  • #4
Think of whether 1 and 5 are true for all invertible 1x1 matrices (i.e., the non-zero reals or non-zero complex numbers).

For 4, expand the left-hand side.

For 6, what are the conditions that a matrix B be self-similar with respect to A?
 

1. What is the inverse of a matrix?

The inverse of a matrix is a matrix that when multiplied with the original matrix produces the identity matrix. It can be thought of as the "opposite" of the original matrix.

2. What are the properties of a matrix inverse?

The properties of a matrix inverse include: 1) the inverse of an invertible matrix is unique, 2) the inverse of a product of matrices is equal to the product of the inverses in reverse order, 3) the inverse of a transpose matrix is equal to the transpose of the inverse, and 4) the inverse of the identity matrix is itself.

3. How is the inverse of a matrix calculated?

The inverse of a matrix can be calculated using various methods, such as Gaussian elimination, the adjugate matrix method, or the elementary row operations method. These methods involve manipulating the matrix to reduce it to its inverse form.

4. Can all matrices have an inverse?

No, not all matrices have an inverse. Only square matrices (same number of rows and columns) that are invertible or non-singular have an inverse. Matrices that are not square, or are singular, do not have an inverse.

5. What is the significance of matrix inverse in mathematics and science?

The inverse of a matrix has various applications in mathematics and science. It is used in solving systems of linear equations, finding solutions to optimization problems, and in data analysis and statistics. It is also important in applications such as computer graphics, cryptography, and physics.

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