What Steps Are Missing in My 4x4 Matrix Inversion Process?

[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 .

 

[HallsofIvy]

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.

 

[M98Ranger]

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.