83. Adjugate

  1. Compute Minor

For each entry of , take the determinant of the submatrix that remains when row and column are removed

  1. Compute Cofactor

Multiply the minor by a sign depending on position

  1. Adjugate (Adjoint)

Take the transpose:

Example

Step 1. Compute minors and cofactors

  • Entry (1, 1): remove row 1 and column 1

Minor:

Cofactor:

Cofactor matrix

  • Entry (1, 2): remove row 1 and column 2

Minor:

Cofactor:

Cofactor matrix

  • Entry (1, 3): remove row 1 and column 3

Minor:

Cofactor:

Cofactor matrix

  • Entry (2, 1): remove row 2 and column 1

Minor:

Cofactor:

Cofactor matrix

  • Entry (2, 2): remove row 2 and column 2

Minor:

Cofactor:

Cofactor matrix

  • Entry (2, 3): remove row 2 and column 3

Minor:

Cofactor:

Cofactor matrix

  • Entry (3, 1): remove row 3 and column 1

Minor:

Cofactor:

Cofactor matrix

  • Entry (3, 2): remove row 3 and column 2

Minor:

Cofactor:

Cofactor matrix

  • Entry (3, 3): remove row 3 and column 3

Minor:

Cofactor:

Cofactor matrix

Step 2. Cofactor matrix

Step 3. Transpose to get adjugate

Relation to inverse:

Relation to identity: