Finding Symmetric and Skew-Symmetric Matrixes B and C for A=B+C

  • Thread starter ashina14
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In summary, a matrix is symmetric if it is equal to its transpose, and skew-symmetric if it is equal to the negative of its transpose. To find a symmetric matrix B and a skew-symmetric matrix C that add up to a given matrix A, one can use the equations A = B + C and A = B^T - C^T. By rearranging these equations and using the property (A+B)^T = A^T + B^T, one can solve for either B or C and then find the other by substitution. A + A^T is symmetric and A - A^T is skew-symmetric, and by adding these two equations together, one can get the original matrix A.
  • #1
ashina14
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Homework Statement




A matrix B is symmetric if B=transpose B.

A matrix C is skew-symmetric if C=−transpose C.

Let A be the matrix given by

A=[[3,6],[-2,1]]

Determine any symmetric matrix B and any skew-symmetric C such that A=B+C

Homework Equations



All given above. Don't know if I need any more.

3. My attempt and issue

I think two equations can be made A =B+C and A = BT - CT
Don't know how to solve them
 
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  • #2
It's been awhile since I've done matrices, but I'll give it a shot. I believe your two equations are correct, you just need to solve for B and C.

Here is what I would do: Solve for either B or C in the first equation ( [itex]A = B + C[/itex] ), plug it into the second equation and use this property: [itex](A+B)^{T}=A^{T}+B^{T}[/itex]. See if that gets you anywhere!
 
  • #3
A+A^T is symmetric, and A-A^T is antisymmetic, yes? Tell me why? Can you think of some way to combine them to get A?
 

FAQ: Finding Symmetric and Skew-Symmetric Matrixes B and C for A=B+C

1. How do I find matrix B?

To find matrix B, you first need to determine the dimensions of the matrix. Then, use the appropriate formula or method to calculate the values for each element in the matrix. This may involve performing operations such as addition, subtraction, multiplication, or division.

2. What is the purpose of finding matrix B?

The purpose of finding matrix B is to represent a set of linear equations or transformations in a concise manner. This can help with solving systems of equations, performing geometric transformations, or analyzing data in a structured way.

3. How do I know if I have found the correct matrix B?

You can verify that you have found the correct matrix B by performing a matrix multiplication with another matrix or performing the operations represented by the matrix and comparing the results to the original data or equation.

4. Can I use any method to find matrix B?

There are several methods for finding matrix B, such as Gaussian elimination, Cramer's rule, or using a calculator or software program. The method you choose may depend on the complexity of the matrix or your personal preference.

5. How is matrix C different from matrix B?

Matrix C may have different dimensions or different values for its elements compared to matrix B. It may also represent a different set of equations or transformations. Additionally, the operations used to find matrix C may be different from those used to find matrix B.

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