How to Find the General Solution for a Matrix with a Free Variable?

In summary, the general solution for the given matrix is [x1, 0, 0, 0] where x1 can be any real number. This represents the null space of the matrix and is spanned by [0, 0, 0, 1].
  • #1
Ad123q
19
0
I was just wondering, if I had a matrix in reduced-row echelon form, say,

1 0 0 2
0 1 3 1
0 0 0 0

then I could write the general solution as a1= -2a4 , a2= -3a3-a4, with a3 and a4 defined in terms of these. (I obtained this solution by putting a1, a2, a3 and a4 under the first, second, third and fourth columns, equating the matrix to zero, and solving for the unknowns under the leading 1's).

But how would I find the general solution of, say

0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0 ?

Any help much appreciated.
 
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  • #2
If you are having a hard time deciphering the matrix representation you can always put them back in equations

In the latter case:

0a + b + 0c + 0d = 0
Thus b = 0

0a + 0b + c + 0d = 0
Thus c = 0

0a + 0b + 0c + d = 0

and 0 = 0 for the last equation

Thus the only vectors to satisfy the matrix are:

[x, 0, 0, 0] transpose. Where x is any real number (I assume these are matrices over R)
 
  • #3
Thanks, so if I was to write x1, x2, x3, x4 under the matrix in question, would the solution space for the general solution just be x = {x1 x2 x3 x4} = {x1 0 0 0} ?
 
  • #4
Ad123q said:
But how would I find the general solution of, say

0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0 ?
I assume the last column is the RHS of a system of linear equations. Note that the last entry of the third row of the matrix is non-0 whereas all the coefficients of the unknowns you are solving for are 0. What does that tell you?
 
  • #5
It would have helped a lot if you had asked the question correctly! There is no such thing as a "general solution of a matrix" any more than there is a general solution of a number.

What you were asking for is a general solution to the equation Ax= 0, or, equivalently, the null space of A.
The matrix equation
[tex]\left[\begin{array}{cccc}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 &1\\ 0 & 0 & 0 & 0\end{array}\right]\left[\begin{array}{c}x_1 \\ x_2\\ x_3\\ x_4\end{array}\right]= \left]\begin{array}{c}0 \\ 0 \\ 0 \\ 0\end{array}\right][/tex]
reduces, as you say, to x1= 0, x2= 0, x3= 0 and 0= 0. Obviously, x1, x2, and x3 must all be 0 but there is no restriction at all on x4. The "general solution" is [0 0 0 x] where x can be any number. The null space is the subspace of R4 spanned by [0 0 0 1].
 
  • #6
Almost right, except I think you got your matrix multiplication backwards; as said before, x2, x3, and x4 must be zero, and there is no restriction on x1.

:)
 
  • #7
Correct you write your matrix in terms of the free variable which in this case would be x1.

So [1,0,0,0] would be the solution set. Also if this is an augmented matrix the general solution would be inconsistent since, 0x1 + 0x2 + 0x3 = 1, cannot be true.
 

1. What is a general solution of a matrix?

A general solution of a matrix is a set of values that satisfies the equation Ax = b, where A is a coefficient matrix, x is a vector of variables, and b is a vector of constants.

2. How is a general solution of a matrix different from a specific solution?

A specific solution is a set of values that satisfies the equation Ax = b for a particular set of values for A and b, while a general solution includes all possible solutions for any set of values for A and b.

3. Can a matrix have more than one general solution?

Yes, a matrix can have infinitely many general solutions because a system of linear equations can have multiple solutions that satisfy the equation Ax = b.

4. How is a general solution of a matrix useful in real-world applications?

A general solution of a matrix is useful in real-world applications for solving systems of equations, such as in engineering or economics problems, where multiple variables are involved.

5. What is the process for finding a general solution of a matrix?

The process for finding a general solution of a matrix involves using row operations to reduce the matrix into its reduced row echelon form, which reveals the general solution in terms of basic variables and free variables.

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