Finding k to Make Matrix Singular: Solving the Mystery

In summary, the conversation is about finding a value k that will make a given matrix singular. The speaker first explains what it means for a matrix to be singular and then shares the matrix they are working with. They then proceed to find a minor and calculate its determinant, using it to solve for k. However, this solution turns out to be incorrect and the other participant helps them realize their mistake. They then successfully calculate the determinant of the entire matrix and find the value of k that makes it zero.
  • #1
mr_coffee
1,629
1
Hello everyone I'm suppose to find a k, that will make this matrix singular, well i looked up what that ment, and it means i have to make the determinant 0. So i the matrix is:
-3 2 1
-9 8 3
(9+k) 2 0

so i found a random minor, say M13..
so i would have
-9 8
(9+k) 2

det = (-9)(2) - [8(9+k)] = -18 -72 -8k = -90 -8k;
so i want k to be 0, so i solve for k
-90 -8k = 0;
k = -90/8, i submitted it, and it was incorrect, any ideas why?
 
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  • #2
Why would making one of the minors 0 make the whole determinant 0? Did you even calculuate the determinant of your matrix? Basically, what you've done is taken:

a*det(A) - b*det(B) + c*det(C)

and said that if as long as one of det(A), det(B), or det(C) is zero, the whole thing is somehow zero. Clearly that's wrong. If the determinants of all the minors were zero, then you'd be fine, but you can't just show one of them to be zero and think that it does anything.

A matrix is singular when its rows and/or columns are linearly dependent. The easiest way to do this is to make two rows or two columns scalar multiples of each other.
 
  • #3
You need to work out the determinant of the entire matrix, not just one of its minors.
 
  • #4
The determinant to be zero is the whole determinant. The det of A.

Work out the det of A with the elements given and equate that to zero.
 
  • #5
ohh my bad, i don't know why I thought that, too much coffee. Thanks it worked out great!
 

1. What is the significance of finding k to make a matrix singular?

Finding k to make a matrix singular is important because it helps identify the critical point at which the matrix becomes singular and has no unique solution. This value of k can provide insights into the behavior of the matrix and its underlying system.

2. How do you determine the value of k to make a matrix singular?

The value of k can be determined by setting the determinant of the matrix equal to zero and solving for k. This will give the critical value at which the matrix becomes singular.

3. Can the value of k differ for different matrices?

Yes, the value of k to make a matrix singular can vary for different matrices. It depends on the specific values and structure of the matrix. Some matrices may not have a value of k that makes them singular.

4. Are there any real-world applications for finding k to make a matrix singular?

Yes, finding k to make a matrix singular can be useful in various fields such as engineering, physics, and economics. It can help in analyzing systems that exhibit sudden changes or instability when a certain parameter reaches a critical value.

5. Are there any alternative methods for solving the mystery of finding k to make a matrix singular?

Yes, there are alternative methods such as using linear algebra techniques, eigenvalue analysis, or computer algorithms to find the value of k that makes a matrix singular. However, the determinant method is the most commonly used and straightforward approach.

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