Row reduced matrix has coefficents

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In summary, the first question deals with finding solutions for Ax=0 when A is row equivalent to the given matrix. The given matrix does not have any solutions because the third and fourth rows contradict each other. The second question is related to the Fredholm Alternative, which states that if A is non-singular, then Ax=b has a unique solution. However, if A is singular and Ax=y has no solution, it is possible for there to exist a vector z such that Ax=z has an infinite number of solutions, but there cannot exist a vector z such that Ax=z has a unique solution.
  • #1
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I just want to confirm these two questions. Thanks in advance.

(1) Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix.

[tex]\left(\begin{array}{uvwxyz}1 & 5 & 2 & -6 & 9 & 0 \\0 & 0 & 1 & -7 & 4 & -8\\0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0\end{array}\right)[/tex]

There are no solutions because row 3 and 4 contradict each other. Row 3 implies no solution.


(2) Suppose A is a 3x3 matrix and y is a vector in R^{3} such that the equation Ax = y does not have a solution. Does there exist a vector z in R^{3} such that the equation Ax = z has a unique solution?

I said no because if the vector y does not have a solution in R^{3}, then this implies the last row of the row reduced matrix has coefficents that are all zero. Therefore, it either has no solution or an infinite number of solutions.
 
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  • #2
Rows three and four do not contradict themselves. All it says is that, with the obvious notation x_6=0 from row 3, and, from row 4, that 0=0. Row 4 contradicts nothing. Besides which, x=0 is always a solution, or otherwise you are saying that the kernel is empty, and since it is a nonempty subspace...
 
  • #3
Problem 2 is related to the "Fredholm Alternative". The equation Ax= b has a unique solution if A is "non-singular". If A is singular then Ax= b has either no solution or an infinite number of solutions depending on b. In this case, since Ax= y has no solution, it might be (and in fact must be) the case that there exist z such that Ax= z has an infinite number of solutions but there cannot exist z such that Ax= z has a unique solution.
 

1. What is a row reduced matrix?

A row reduced matrix is a matrix that has undergone a series of elementary row operations, resulting in a simplified form where each row has a leading 1 and zeros in all other entries in the same column.

2. What are the coefficients in a row reduced matrix?

The coefficients in a row reduced matrix are the numbers that appear in the matrix and are used to represent the variables in a system of linear equations. These coefficients are typically represented by letters such as a, b, c, etc.

3. How is a matrix row reduced?

A matrix is row reduced by performing a series of elementary row operations, such as swapping rows, multiplying a row by a non-zero constant, or adding a multiple of one row to another row. These operations are performed in a specific order to transform the matrix into its row reduced form.

4. What is the purpose of row reducing a matrix?

The purpose of row reducing a matrix is to simplify a system of linear equations and make it easier to solve. It also helps in identifying any inconsistencies or redundancies in the system of equations. Additionally, row reducing a matrix can also be used to find the rank and nullity of a matrix.

5. Can a row reduced matrix have more than one solution?

Yes, a row reduced matrix can have more than one solution. This is known as an infinite solution, where the system of equations has multiple sets of values that satisfy all the equations. This can occur when there are free variables present in the matrix, which can take on any value.

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