Inverting a 4x4 matrix

  • Thread starter tony blair
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There is a little more to do, but you should be able to use this to finish the problem.In summary, the conversation involves finding the inverse of a matrix and the steps taken to reduce it to row echelon form. The final steps are not provided, but the process includes swapping rows and performing operations such as addition and subtraction to create a matrix with a 1 in the "pivot position" and zeros in the rest of the column.
  • #1
tony blair
here's what I've done
-got transpose of A

|-3 0 0 1 |
| 1 -2 1 0 |
| 2 -3 2 1 |
| 1 2 -1 2 |


the deteminant = 8 therefore (its's invertible)

**the problem***


|-3 0 0 1 | 1 0 0 0 |
| 1 -2 1 0 | 0 1 0 0 |
| 2 -3 2 1 | 0 0 1 0 |
| 1 2 -1 2 | 0 0 0 1 |

i've tried reducing this (even tried using adjoint method)
but i keep getting a different answer

could someone please post for me the row reductions to
get the inverse to this thing i.e
row2 -row3 on row3 etc etc etc .
 
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  • #2
You talked about "reducing this" but didn't say exactly what it is you want to arrive at. If I remember correctly you wanted to find the inverse of a matrix. Was it A or the transpose of A? If you want to find the inverse of the matrix A, I don't see any reason to work with the transpose.

|-3 0 0 1 | 1 0 0 0 |
| 1 -2 1 0 | 0 1 0 0 |
| 2 -3 2 1 | 0 0 1 0 |
| 1 2 -1 2 | 0 0 0 1 |

If I were going to do this, the first thing I would do is swap the first two rows so I will have a 1 in the first row of the first column. Then (1) add 3 times that (new) first row to the (new) second row, (2) subtract 2 times that (new) first row from the third row, and (3) Subtract that (new) first row from the fourth row to get:
[1 -2 1 0| 0 1 0 0 ]
[0 -6 3 1| 1 3 0 0 ]
[0 1-2 1| 0-2 1 0 ]
[0 4 2 2| 0-1 0 1 ]

Now swap the second and third rows to get a 1 in the "pivot position" (second row, second column). Add twice that new second row to the first row, add 6 time the new second row to the third row, and subtract 4 times that new second row to the fourth row to get:

OOPs, got to run! (Going to see "The Barber of Seville"!) You'll have to work it out yourself.
 
  • #3



Inverting a matrix is a complex process and requires careful calculations and steps. Here are the steps to invert a 4x4 matrix:

Step 1: Find the determinant of the matrix. You have already calculated the determinant to be 8, which means the matrix is invertible.

Step 2: Find the cofactors of each element in the matrix. To find the cofactor of an element, you need to find the determinant of the submatrix formed by removing the row and column in which the element is located. For example, the cofactor of the element -3 would be the determinant of the submatrix:

|-2 1 0 |
|-3 2 1 |
| 2 -1 2 |

Step 3: Create the adjoint matrix by replacing each element in the original matrix with its corresponding cofactor, but with the sign flipped for every other element. This means that the first element in the first row of the adjoint matrix would be the cofactor of the element in the first row and first column of the original matrix, but with the sign flipped. The second element in the first row of the adjoint matrix would be the cofactor of the element in the first row and second column of the original matrix, but with the sign not flipped, and so on.

Step 4: Transpose the adjoint matrix. This means that you switch the rows and columns of the matrix.

Step 5: Multiply the transposed adjoint matrix by the reciprocal of the determinant. This will give you the inverse of the original matrix.

I can see that you have tried to use the adjoint method, but it seems like you may have made some mistakes in your calculations. I would recommend checking your work and double-checking your steps to ensure accuracy. It may also be helpful to use a calculator or software to assist with the calculations.
 

What is a 4x4 matrix?

A 4x4 matrix is a rectangular array of numbers with 4 rows and 4 columns. It is commonly used in linear algebra to represent linear transformations, such as rotations, translations, and scaling.

Why do we need to invert a 4x4 matrix?

Inverting a 4x4 matrix is important in solving systems of linear equations and finding the inverse of a linear transformation. It also has applications in computer graphics, robotics, and physics.

How do you invert a 4x4 matrix?

To invert a 4x4 matrix, you can use various methods such as Gaussian elimination, LU decomposition, or the adjugate matrix method. These methods involve performing a series of mathematical operations on the matrix to find its inverse.

What is the purpose of finding the inverse of a 4x4 matrix?

The inverse of a 4x4 matrix can be used to reverse the effects of a linear transformation. It can also be used to solve systems of linear equations, find the determinant of the original matrix, and perform other mathematical calculations.

Are there any limitations to inverting a 4x4 matrix?

Yes, there are limitations to inverting a 4x4 matrix. One limitation is that not all 4x4 matrices are invertible. This means that some matrices do not have an inverse, which can happen when the determinant of the matrix is equal to 0.

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