If the Determinant is 0, what all is true? matrices fun

In summary, Carl is asking for help with a problem involving a general matrix equation. There are five options to consider, including A, B, C, D, and E. Carl believes that A and B are incorrect and that the correct answer must be a combination of C, D, and E. However, Carl has tried various combinations of C, D, and E and none seem to work. Another person, named Carl, is trying to understand Carl's reasoning for dismissing option A.
  • #1
mr_coffee
1,629
1
Hello everyone I'm stuck on this problem, how its wrong, i got parts a and c, but b i can't get. Hre is what it says:

Consider the following general matrix equation:
[a1 a2] =
|m11 m12| * [x1 x2]
|m21 m22|

which can also be written as A = MX
Determinant of M = m11m22 - m12m21

A. given any X there is one and only one A which will satisfy the equation.
B. given any A there is one and only one X which will satisfy the equation.
C. there is no value of Xwhich satisfies the equation when A =0 .
D. some values of A will have no values of X which will satisfy the equation.
E. some values of A (such as A=0) will allow more than one X to satisfy the equation.

I said for sure A, and B are wrong because they are what happens if Det(M) is not equal to 0. so it has to be a combination of C,D, and E.

I tried like all combinations of C,D,and E, and nothing works!
 
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  • #2
mr_coffee said:
I said for sure A, and B are wrong ...

You might go over your logic behind why you think that A is wrong.

Carl
 

FAQ: If the Determinant is 0, what all is true? matrices fun

1. What does it mean if the determinant of a matrix is 0?

If the determinant of a matrix is 0, it means that the matrix is not invertible. This means that the matrix cannot be transformed back into its original form using elementary row operations, and it also means that there is no unique solution to a system of linear equations using this matrix.

2. Can a matrix have a determinant of 0 if it has non-zero entries?

Yes, a matrix can have a determinant of 0 even if it has non-zero entries. The determinant is a measure of the linear independence of the vectors that make up the columns or rows of a matrix, so even if the entries are non-zero, if they are linearly dependent, the determinant will be 0.

3. How does the determinant being 0 affect the eigenvalues of a matrix?

If the determinant of a matrix is 0, it means that the matrix is not invertible, which also means that the matrix has at least one eigenvalue of 0. This is because the eigenvalues of a matrix are the solutions to the characteristic equation, and if the matrix is not invertible, the characteristic equation will have at least one solution of 0.

4. Is it possible for a matrix to have a determinant of 0 if it is a square matrix?

Yes, it is possible for a square matrix to have a determinant of 0. The determinant of a matrix is only defined for square matrices, and even though they have the same number of rows and columns, the determinant can still be 0 if the matrix is not invertible.

5. Can the determinant of a matrix ever be negative?

No, the determinant of a matrix can never be negative. The determinant is a scalar value, so it cannot have a negative sign. It can be 0 or positive, but never negative.

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