Techniques for solving type of Matrix problems

In summary, the conversation discusses the sizes of matrices A, B, and C in various scenarios. In 1a, it is determined that B is a 2x2 matrix and C is a 2x2 matrix. In 1b, it is found that B is a 3x3 matrix and C is a 2x2 matrix. In 2a, it is concluded that B is a 5x5 matrix and C is a 2x5 matrix. In 2b, the sizes of A, B, and C are determined to be 6x5, 5x7, and 7x7 respectively. To find these sizes, various techniques can be used, such as
  • #1
niteshadw
20
0
1)
a)
If A =
1 2
0 3
and B is an upper-triangular matrix such that tr(B) = 0 and
AB =
1 -1
0 -3
then B = _____

AND
b)
If A =
1 5
-1 3
and A = B+C where B is symmetric and C is skew-symmetric, then
B = ___ and C = ____.

2)
a)
If A, B and C are matrices such that A^TB^(-1)C is a column matrix, and A is a 2x5 matrix, then the size of B is _____ and the size of C is ___.

b)
If B^(−1)A^TBC is a 6 × 7 matrix, then the size of A is ,
the size of B is ___, and the size of C is ____.


Are there some easy techniques that can be used to find the sizes of each of the matrix, such as 2a and 2b? I kind of have an idea of how to do those mentioned in 1a but 1b having a bit trouble. These are not homework questions but questions from old exams. I have an exam coming up and I'm trying to review. Any suggestions would be much appreciated. Thank you
 
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  • #2
!1a) B =0 00 0C = 0 -11 01b) B = 3 22 4C = 0 1-1 02a) The size of B is 5x5 and the size of C is 2x5.2b) The size of A is 6x5, the size of B is 5x7, and the size of C is 7x7.
 
  • #3


1a) To solve this problem, we can use the fact that the trace of a matrix is equal to the sum of its diagonal elements. Since tr(B) = 0, we know that the sum of the diagonal elements of B must be 0. Using this information, we can set up the following equation:

1 + (-3) = 0

This means that the diagonal elements of B are 1 and -3. Since B is an upper-triangular matrix, the only possible values for the diagonal elements are 1 and -3. Therefore, B must be:

B =
1 0
0 -3

b) To solve this problem, we can use the fact that a symmetric matrix is equal to its transpose, while a skew-symmetric matrix is equal to the negative of its transpose. Using this information, we can set up the following equations:

B = B^T
C = -C^T

We can then solve for B and C by setting up a system of equations using the given values for A and the equations above. This will result in:

B =
1 2
2 3

C =
0 -7
7 0

2a) To solve this problem, we can use the fact that the size of a matrix is given by the number of rows and columns it has. Since A is a 2x5 matrix, it has 2 rows and 5 columns. We also know that B^(-1)C is a column matrix, which means it has only one column. Therefore, the size of B must be 5x1 and the size of C must be 5x1.

2b) Similar to 2a, we can use the fact that the size of a matrix is given by the number of rows and columns it has. Since B^(-1)A^TBC is a 6x7 matrix, we know that B^(-1)A^T must be a 6x7 matrix as well. Therefore, the size of A must be 7x6, the size of B must be 7x7, and the size of C must be 7x7.
 

1. What are the different types of matrix problems that can be solved using techniques?

There are several types of matrix problems that can be solved using different techniques. Some common types include systems of linear equations, determinants, eigenvalues and eigenvectors, and matrix factorization.

2. What is the Gauss-Jordan elimination method and how is it used to solve matrix problems?

The Gauss-Jordan elimination method is a technique used to solve systems of linear equations by transforming the system into an equivalent system with a simpler form. This method involves performing elementary row operations on the augmented matrix until it is in reduced row-echelon form, which can then be used to find the solutions to the system.

3. Can matrix problems be solved using software or do they require manual calculations?

Matrix problems can be solved using software such as calculators or computer programs, but they can also be solved manually using various techniques. Using software can save time and reduce the chances of errors, but it is important to understand the underlying concepts and techniques used to solve the problems.

4. How can matrix problems be applied in real-world situations?

Matrix problems have various applications in real-world situations, such as in engineering, physics, economics, and computer graphics. They can be used to model and solve systems of equations, analyze data, and optimize solutions in various fields.

5. Are there any common mistakes to avoid when solving matrix problems?

One common mistake to avoid when solving matrix problems is not paying attention to the order of operations when performing calculations. It is important to carefully follow the steps and rules of the chosen technique to avoid errors. Additionally, it is important to check the final solutions to ensure they make sense in the context of the problem.

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