How Does Subtracting c from ann Affect Matrix Singularity?

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If they are not equal, then the matrix B is singular, which means that subtracting c from ann makes the resulting matrix singular. In summary, by replacing det(A) with the relevant equation and multiplying both sides by Ann, we can show that subtracting c from ann results in a matrix with a determinant of 0. This can be proven by defining a matrix B with the (n,n) element as a_{nn} - c and calculating det(B). If the resulting determinant is not equal to det(A), then the matrix B is singular, proving that subtracting c from ann results in a singular matrix.
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Ylle
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Homework Statement


Let A be a non-singular n x n matrix with a non-zero cofactor Ann and let

c = det(A) / Ann

Show that if we subtract c from ann, then the resulting matrix will be singular.


Homework Equations



det(A) = a1nA1n+...+annAnn


The Attempt at a Solution



Well, if I replace det(A) with the one in "Relevant eq.", and multiply both sides with Ann I get:

cAnn = a1nA1n+...+annAnn

Then if I subtract cAnn from both sides i get:

0 = a1nA1n+...+annAnn - cAnn
, which we can rewrite to:

0 = a1nA1n+...+(ann-c)Ann


And now I'm not sure if I'm done ?

It seems like I need to define another matrix of some sort to define det(B) = 0.
But I'm not quite sure how I do that. Can anyone give me a hint ? :)
 
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  • #2
Ylle said:

Homework Statement


Let A be a non-singular n x n matrix with a non-zero cofactor Ann and let

c = det(A) / Ann

Show that if we subtract c from ann, then the resulting matrix will be singular.

Homework Equations



det(A) = a1nA1n+...+annAnn

The Attempt at a Solution



Well, if I replace det(A) with the one in "Relevant eq.", and multiply both sides with Ann I get:

cAnn = a1nA1n+...+annAnn

Then if I subtract cAnn from both sides i get:

0 = a1nA1n+...+annAnn - cAnn
, which we can rewrite to:

0 = a1nA1n+...+(ann-c)AnnAnd now I'm not sure if I'm done ?

It seems like I need to define another matrix of some sort to define det(B) = 0.
But I'm not quite sure how I do that. Can anyone give me a hint ? :)

Yes, define a matrix B which is exactly the same as A except for the (n,n) element, which will be

[tex]a_{nn} - c[/tex] instead of [tex]a_{nn}[/tex]

Then calculate det(B) the same way you calculated det(A), and compare the answers you get.
 

1. What is a matrix singular problem?

A matrix singular problem refers to a situation where a matrix is unable to be inverted because it does not have an inverse. This can happen when the determinant of the matrix is equal to zero, making the matrix singular.

2. Why is a matrix singular problem important?

A matrix singular problem is important because it can affect the validity and accuracy of mathematical calculations involving the matrix. It can also lead to errors in computer programs that use matrices for data manipulation.

3. How can a matrix singular problem be avoided?

A matrix singular problem can be avoided by ensuring that the matrix being used is non-singular, meaning it has an inverse. This can be achieved by checking the determinant of the matrix before using it in calculations.

4. What are some applications of matrices that can encounter singular problems?

Matrices are used in various fields such as engineering, physics, economics, and computer science. Some common applications that can encounter singular problems include solving linear systems of equations, data compression, and image processing.

5. How can a matrix singular problem be solved?

A matrix singular problem can be solved by using specialized methods such as the pseudo-inverse or the generalized inverse. These methods allow for the calculation of a pseudo-inverse matrix for a singular matrix, which can then be used in place of the original matrix in calculations.

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