Find inverse matrix using determinants and adjoints

In summary, the conversation was about the process of finding the inverse of a given matrix using determinants and adjoints. The individual seeking help was having trouble with the calculation and was unsure if they were doing something wrong with the cofactors. After discussing the process and verifying the calculation of the determinant, it was determined that the individual had made a mistake with the sign, which led to a wrong answer. The conversation ended with the individual thanking everyone for their help and successfully finding the inverse matrix by correcting the determinant of the initial matrix.
  • #1
ducmod
86
0
Hello!

Please, help me to see my mistake - for quite a while I can't solve a very easy matrix.

I have to
find the inverse of the given matrix using their determinants and adjoints.
4 6 -3
3 4 -3
1 2 6

to find adjoint matrix I need to find cofactors 11, 12, etc till 33.
Cofactor11 = (-1)^(1+1) x det([4 -3] => C11 = 24 + 6 = 30
2 6
Cofactor12 = (-1)^(1+2) x det([3 -3] => C12 = -(18 + 3) = -21
1 6

Cofactor13 = (-1)^(1+3) x det([3 4] => C13 = 6 - 4 = 2
1 2

Cofactor21 = (-1)^(2+1) x det([6 -3] => C21 = -(36 + 6) = -42
2 6

Cofactor22 = (-1)^(2+2) x det([4 -3] => C22 = 24 + 3 = 27
1 6
Cofactor23 = (-1)^(2+3) x det([4 6] => C23 = -(8 - 6) = -2
1 2

Cofactor31 = (-1)^(3+1) x det([6 -3] => C31 = -6
4 -3

Cofactor32 = (-1)^(3+2) x det([4 -3] => C32 = -(-12 + 9) = 3
3 -3
Cofactor33 = (-1)^(3+3) x det([4 6] => C33 = 16 - 18 = -2
3 4

Adjoint matrix:
30 -42 -6
-21 27 3
2 -2 -2

det(of initial matrix taken by the first row) = 4 x (-1)^(1+1) x det([A11)] + 6 x (-1)^(1+2) x det([A12)] + (-3) x (-1)^(1+3) x det([A13)] = 4 x 30 + 6 x (-21) + (-3) x (-2) = 0

and if I try to find the det of intial matrix by expanding over the third row I get 1 x (-6) + 2 x (-1) x (-3) + 6 x (-2) = -2

and if I try to find the det of intial matrix by expanding over the second row I get (-3) x 42 + 4 x 27 + 3 x 2 = -12

I have tried multiple times, with different rows for initial matrix, and each time I get a different result.
Am I doing something wrong with cofactors?

Thank you!
 
Last edited:
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  • #3
ducmod said:
Hello!

Please, help me to see my mistake - for quite a while I can't solve a very easy matrix.

I have to
find the inverse of the given matrix using their determinants and adjoints.
4 6 -3
3 4 -3
1 2 6

to find adjoint matrix I need to find cofactors 11, 12, etc till 33.
Cofactor11 = (-1)^(1+1) x det([4 -3] => C11 = 24 + 6 = 30
2 6
Cofactor12 = (-1)^(1+2) x det([3 -3] => C12 = -(18 + 3) = -21
1 6

Cofactor13 = (-1)^(1+3) x det([3 4] => C13 = 6 - 4 = 2
1 2

Cofactor21 = (-1)^(2+1) x det([6 -3] => C21 = -(36 + 6) = -42
2 6

Cofactor22 = (-1)^(2+2) x det([4 -3] => C22 = 24 + 3 = 27
1 6
Cofactor23 = (-1)^(2+3) x det([4 6] => C23 = -(8 - 6) = -2
1 2

Cofactor31 = (-1)^(3+1) x det([6 -3] => C31 = -6
4 -3

Cofactor32 = (-1)^(3+2) x det([4 -3] => C32 = -(-12 + 9) = 3
3 -3
Cofactor33 = (-1)^(3+3) x det([4 6] => C33 = 16 - 18 = -2
3 4

Adjoint matrix:
30 -42 -6
-21 27 3
2 -2 -2

det(of initial matrix taken by the first row) = 4 x (-1)^(1+1) x det([A11)] + 6 x (-1)^(1+2) x det([A12)] + (-3) x (-1)^(1+3) x det([A13)] = 4 x 30 + 6 x (-21) + (-3) x (-2) = 0

I have tried multiple times, with different rows for initial matrix, and each time I get a different result.
Am I doing something wrong with cofactors?

Thank you!
Your calculation of the det(A) = 0 is incorrect. If A has a zero determinant, can it have an inverse?
 
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  • #4
The last sign you typed in the determinant calculation is wrong.
 
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  • #5
Consider that you are in fact in front of three systems with each three linear combinations depending on three unknown scalars; for example, the system corresponding to the first column is:

4 . x11 + 6 . x12 - 3 . x13 = 1

3 . x11 + 4 . x12 - 3 . x13 = 0

1 . x11 + 2 . x12 + 6 . x13 = 0

First verify that the discriminant of the system doesn’t vanish; that is let calculate:

+ 4 . [4 . 6 – 2 . (-3)] – (+6) . [3 . 6 – 1 . (-3)] + (-3) . [3 . 2 – 1 . 4]

=

4 . (24 + 6) – 6 . (18 + 3) + 3 . (6 – 4)

=

4 . 30 – 6 . 21 + 3 . 2

=

120 – 126 + 6

=

0

Unfortunately, yes that discriminant vanishes. The matrix cannot be inverted. The most important is to know how a discriminant of a system /determinant of a matrix must be calculated.
 
  • #6
Blackforest said:
Consider that you are in fact in front of three systems with each three linear combinations depending on three unknown scalars; for example, the system corresponding to the first column is:

4 . x11 + 6 . x12 - 3 . x13 = 1

3 . x11 + 4 . x12 - 3 . x13 = 0

1 . x11 + 2 . x12 + 6 . x13 = 0

First verify that the discriminant of the system doesn’t vanish; that is let calculate:

+ 4 . [4 . 6 – 2 . (-3)] – (+6) . [3 . 6 – 1 . (-3)] + (-3) . [3 . 2 – 1 . 4]

=

4 . (24 + 6) – 6 . (18 + 3) + 3 . (6 – 4)
You flipped a sign here. The matrix can be inverted.
 
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  • #7
Thank you everyone! I see - my mistake is in the sign, as you have pointed out. Expanding by the first and second row both produce -12.
 
  • #8
Once again - thank you very much for your help! I have found the inverse matrix by correcting the determinant of the initial matrix.
 

Related to Find inverse matrix using determinants and adjoints

1. What is the purpose of finding the inverse matrix using determinants and adjoints?

The purpose of finding the inverse matrix using determinants and adjoints is to solve systems of linear equations and to perform transformations in linear algebra. It is also useful in solving problems in physics, economics, and engineering.

2. How do you find the inverse matrix using determinants and adjoints?

To find the inverse matrix using determinants and adjoints, you first need to calculate the determinant of the original matrix. Then, find the adjoint matrix by taking the transpose of the cofactor matrix. Finally, divide the adjoint matrix by the determinant to get the inverse matrix.

3. What is the relationship between determinants and inverse matrices?

The determinant of a matrix is a scalar value that represents the scaling factor of the linear transformation represented by the matrix. The inverse matrix, on the other hand, undoes the transformation carried out by the original matrix. The inverse matrix is calculated using determinants and adjoints, and the determinant of the inverse matrix is the reciprocal of the determinant of the original matrix.

4. When is it necessary to find the inverse matrix using determinants and adjoints?

It is necessary to find the inverse matrix using determinants and adjoints when you need to solve systems of linear equations or perform transformations in linear algebra. It is also used in solving optimization problems and in finding the inverse of a matrix in computer graphics.

5. Are there any limitations to finding the inverse matrix using determinants and adjoints?

Yes, there are limitations to finding the inverse matrix using determinants and adjoints. The matrix must be square and non-singular (i.e. have a non-zero determinant). If the matrix is singular, the inverse matrix does not exist. Additionally, the process can be computationally intensive for large matrices, so other methods may be preferred in those cases.

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