Linear algebra question about matrices

In summary, to determine the values of k for which the given matrix is the augmented matrix of a system with infinitely many solutions, one can use row reduction to put the matrix in upper triangular form. If the determinant of the matrix is 0, then the system has infinitely many solutions. To find all values of k for which the matrix has a system with no solutions, one can use the same method and check for a determinant of 0. Additionally, one can multiply rows by appropriate factors to create a diagonal of zeros and solve for k.
  • #1
LaraCroft
14
0
How would one determine the values of k such that the following matrix is the augmented matrix of a system with infinitely many solutions:

[ (k + 2) -2 1 | 2 ]
[ (k + 3) (k+ 3) 2 | 2 ]
[ (k + 2) -2 (k -1 ) | -3 ]

Also, how would I get all values of k such that the matrix for the same to be an augmented matrix of a system with no solutions?

It confuses me on how to find all possible values of k!

Thank you!:smile:
 
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  • #2
LaraCroft said:
How would one determine the values of k such that the following matrix is the augmented matrix of a system with infinitely many solutions:

[ (k + 2) -2 1 | 2 ]
[ (k + 3) (k+ 3) 2 | 2 ]
[ (k + 2) -2 (k -1 ) | -3 ]

Also, how would I get all values of k such that the matrix for the same to be an augmented matrix of a system with no solutions?

It confuses me on how to find all possible values of k!

Thank you!:smile:
Try to row reduce it!
Reducing it to upper triangular form, so that you have only 0s below the main diagonal, you wind up with 0 0 f(k) | g(k). There is not a unique solution if f(k)= 0. There are an infinite number of solutions if g(k)= 0 also. (If f(k)= 0 and g(k) is not 0, then there is no solution.)
 
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  • #3
I would have set the determinant equal to zero. Of course the determinate can be calculated via row reduction.
 
  • #4
The problem with that is that if the determinant is 0, then the matrix equation may have infinitely many solutions or NO solution.
the question here was specifically to determine k so the equation has infinitely many solutions. You need to include the right hand side to determine that.
 
  • #5
HallsofIvy said:
The problem with that is that if the determinant is 0, then the matrix equation may have infinitely many solutions or NO solution.
the question here was specifically to determine k so the equation has infinitely many solutions. You need to include the right hand side to determine that.

You can always substitute the roots obtained by the determinate back into the original system and then do row reduction. You can also put the system into upper triangular form while computing the determinate.
 
  • #6
Ok...

Firstly, thank you everyone for responding...

Secondly, I am still not getting how to determine all the values of K, so that the matrix (call it A) is the augmented matrix of a system with infinitely many solutions. How would I find all values of K?

I understand that I need to row echelon it...but I think the way I am doing it is wrong...since the first entry cannot be 1...I start by doing Row 2 minus Row 1...and I I continue...however I am confused on what to do next in order to find k!

Apparently I should also be able to determine all values of k so that the matrix has a system with no solutions...

Thanks again:smile:
 
  • #7
LaraCroft said:
Ok...
I understand that I need to row echelon it...but I think the way I am doing it is wrong...since the first entry cannot be 1...I start by doing Row 2 minus Row 1...and I I continue...however I am confused on what to do next in order to find k!

Before you do your subtraction multiply rows one and three by (K+3) and row 2 by (k+1)
 

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used to represent and solve systems of linear equations.

2. What is the difference between a row matrix and a column matrix?

A row matrix has only one row, while a column matrix has only one column. In other words, a row matrix is represented by a horizontal line of numbers, and a column matrix is represented by a vertical line of numbers.

3. How do you add or subtract matrices?

To add or subtract matrices, the matrices must have the same dimensions. You simply add or subtract the corresponding elements in each matrix. For example, to add two 2x2 matrices, add the top left elements, then the top right elements, then the bottom left elements, and finally the bottom right elements.

4. Can you multiply any two matrices together?

No, in order for two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

5. How can matrices be used in real-world applications?

Matrices are used in many different fields, including engineering, physics, economics, and computer graphics. They can be used to model and solve various problems, such as finding the equilibrium point in an economic system, predicting the motion of a system in physics, or creating computer-generated images in video games and movies.

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