Inverse of a Matrix
using Minors, Cofactors and Adjugate

Note: besides check out Matrix Inverse by Row Operations and the Matrix Calculator

Nosotros tin can calculate the Changed of a Matrix by:

  • Stride 1: computing the Matrix of Minors,
  • Pace two: and so turn that into the Matrix of Cofactors,
  • Step 3: and then the Adjugate, and
  • Step 4: multiply that by 1/Determinant.

But information technology is all-time explained past working through an case!

Example: observe the Inverse of A:

It needs 4 steps. It is all elementary arithmetic merely there is a lot of it, then try not to brand a mistake!

Step 1: Matrix of Minors

The first step is to create a "Matrix of Minors". This footstep has the nearly calculations.

For each element of the matrix:

  • ignore the values on the current row and column
  • calculate the determinant of the remaining values

Put those determinants into a matrix (the "Matrix of Minors")

Determinant

For a 2×two matrix (ii rows and 2 columns) the determinant is easy: ad-bc

Call back of a cantankerous:

  • Blueish means positive (+ad),
  • Red means negative (-bc)
 A Matrix

(It gets harder for a 3×3 matrix, etc)

The Calculations

Hither are the showtime 2, and last two, calculations of the "Matrix of Minors" (notice how I ignore the values in the current row and columns, and summate the determinant using the remaining values):

matrix of minors calculation steps

And hither is the calculation for the whole matrix:

matrix minors result

Step 2: Matrix of Cofactors

checkerboard of plus and minus

This is easy! Just apply a "checkerboard" of minuses to the "Matrix of Minors". In other words, we demand to change the sign of alternate cells, like this:

matrix of cofactors

Stride three: Adjugate (also called Adjoint)

Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the aforementioned):

matrix adjugate

Pace 4: Multiply by 1/Determinant

Now find the determinant of the original matrix. This isn't too hard, because we already calculated the determinants of the smaller parts when we did "Matrix of Minors".

A Matrix

Using:

Elements of tiptop row: iii, 0, ii
Minors for pinnacle row: two, two, 2

We end up with this calculation:

Determinant = iii×2 − 0×2 + ii×2 = x

Note: a small simplification is to multiply by the cofactors (which already accept the "+−+−" pattern), and and so we only add each time:

Determinant = 3×ii + 0×(−2) + 2×2 = ten

Your Plow: try this for any other row or cavalcade, y'all should also get 10.

At present we multiply the Adjugate past 1/Determinant to go:

matrix adjugate by 1/det gives inverse

And we are done!

Compare this answer with the ane we got on Inverse of a Matrix using Elementary Row Operations. Is information technology the same? Which method exercise y'all prefer?

Larger Matrices

It is exactly the same steps for larger matrices (such as a 4×4, five×5, etc), only wow! at that place is a lot of adding involved.

For a 4×4 Matrix nosotros have to summate 16 3×3 determinants. And then it is often easier to apply computers (such as the Matrix Calculator.)

Conclusion

  • For each chemical element, summate the determinant of the values not on the row or column, to make the Matrix of Minors
  • Apply a checkerboard of minuses to make the Matrix of Cofactors
  • Transpose to brand the Adjugate
  • Multiply by 1/Determinant to make the Inverse

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