Simple Linear Algebra (determinant invertibility)

In summary, a good book for algebra (group theory) that can be helpful for a math entrance exam is "A First Course in Abstract Algebra" by John B. Fraleigh. Other recommended books include "The Theory of Groups" by Marshall Hall and "An Introduction to the Theory of Groups" by Joseph J. Rotman. It is important to note that there is not a single "correct" way to solve a proof, as long as the logic is sound and the conclusion is correct.
  • #1
slashrulez
7
0
I have a quick question about whether or not a matrix is invertible. The question asked is pretty simple, "Suppose that A is a square matrix such that det(A^4) = 0. Show that A cannot be invertible." I know how to explain it, but I'm not sure if it's really the "correct" way, as in I'm not missing anything or making assumptions I wouldn't be allowed to make on an exam.

So my stab at it:

det(A^4) = det(AAAA) = detA detA detA detA = 0, therefore det A = 0

If A is invertible, there exists a B such that

AB = BA = I
det(AB) = det(I)
detA detB = 1
Therefore, for the matrix to be invertible, detA must be non-zero, which it isn't

Like, that seems right to me, but I'm not sure if I have to do any additional work for the part with the inverse to show I understand it.
 
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  • #2
Looks perfect to me.
 
  • #3
slashrulez said:
I have a quick question about whether or not a matrix is invertible. The question asked is pretty simple, "Suppose that A is a square matrix such that det(A^4) = 0. Show that A cannot be invertible." I know how to explain it, but I'm not sure if it's really the "correct" way, as in I'm not missing anything or making assumptions I wouldn't be allowed to make on an exam.

So my stab at it:

det(A^4) = det(AAAA) = detA detA detA detA = 0, therefore det A = 0

If A is invertible, there exists a B such that

AB = BA = I
det(AB) = det(I)
detA detB = 1
Therefore, for the matrix to be invertible, detA must be non-zero, which it isn't

Like, that seems right to me, but I'm not sure if I have to do any additional work for the part with the inverse to show I understand it.

The only thing I would add to the earlier approval is that there is not a single "correct way". You logic is flawless and the conclusion correct, so it is at the least a correct way.

The only real rule for doing a proof is that the logic be correct and that you demonstrate that which was to be demonstrated. There are often many valid proofs of a single theorem. Some are perhaps more elegant than others, but elegance is in the eye of the beholder and quite often the elegant proofs result from reviewing and refining that which was discovered by others.
 
  • #4
cdniki said:
Hi
Can anybody suggest me a good book for algebra(group theory)which can be helpful me for math entrance exam.

I am not quite sure why you would be looking for a book on group theory for an entrance exam. However, Mashall Hall's The Theory of Groups is a classic on that subject and quite good. Rotman's An Introduction to the Theory of Groups is also good.
 
  • #5
If you're posting a homework problem you should make a new thread in the homework section. Read the sticky at the top of the section seeing what you should include (in particular, you should include work that you've done)
 
  • #6
cdniki said:
Hi
Can anybody suggest me a good book for algebra(group theory)which can be helpful me for math entrance exam.
A good introductory book is "A First Course in Abstract Algebra". ISBN 0-201-76390-7
 
  • #7
cdniki's posts were removed as he needs to post in the proper forum and show his work.
 

Related to Simple Linear Algebra (determinant invertibility)

1. What is a determinant in linear algebra?

The determinant is a mathematical concept that is used to determine whether a square matrix is invertible or not. It is a scalar value that is calculated from the elements of a matrix and can provide information about the properties of the matrix.

2. How is the determinant calculated?

The determinant of a 2x2 matrix is calculated by multiplying the elements in the main diagonal and subtracting the product of the elements in the other diagonal. For larger matrices, the determinant can be calculated using different methods such as cofactor expansion or Gaussian elimination.

3. What does it mean for a matrix to be invertible?

A matrix is invertible if it has a unique inverse, meaning that it can be multiplied by another matrix to produce the identity matrix. In other words, the inverse of a matrix cancels out the original matrix when multiplied together.

4. How do I know if a matrix is invertible using its determinant?

A matrix is invertible if and only if its determinant is non-zero. If the determinant is equal to zero, the matrix is not invertible and is considered singular. This means that the matrix does not have a unique solution and cannot be inverted.

5. What is the significance of determinant invertibility in linear algebra?

Determinant invertibility is important in linear algebra because it allows us to determine whether a system of linear equations has a unique solution. If a matrix is invertible, it means that the system has a unique solution and can be solved using matrix operations. Determinant invertibility also plays a role in finding the inverse of a matrix, which is a useful tool in many mathematical and scientific applications.

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