83. Adjugate

adj(𝐴)
  1. Compute Minor 𝑀𝑖𝑗

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

  1. Compute Cofactor 𝐶𝑖𝑗

Multiply the minor by a sign depending on position

𝐶𝑖𝑗=(−1)𝑖+𝑗𝑀𝑖𝑗
  1. Adjugate (Adjoint)

Take the transpose:

adj(𝐴)=𝐶𝑇
Example
𝐴=[123045106]

Step 1. Compute minors and cofactors

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

Minor:

[
]det([4506])=24

Cofactor:

𝐶11=(−1)1+1⋅24=1⋅24=24

Cofactor matrix

𝐶=[24]
  • Entry (1, 2): remove row 1 and column 2

Minor:

[
]det([0516])=−5

Cofactor:

𝐶12=(−1)1+2⋅−5=−1⋅−5=5

Cofactor matrix

𝐶=[245]
  • Entry (1, 3): remove row 1 and column 3

Minor:

[
]det([0410])=−4

Cofactor:

𝐶13=(−1)1+3⋅−4=1⋅−4=−4

Cofactor matrix

𝐶=[245−4]
  • Entry (2, 1): remove row 2 and column 1

Minor:

[
]det([2306])=12

Cofactor:

𝐶21=(−1)2+1⋅12=−1⋅12=−12

Cofactor matrix

𝐶=[245−4−12]
  • Entry (2, 2): remove row 2 and column 2

Minor:

[
]det([1316])=3

Cofactor:

𝐶22=(−1)2+2⋅3=1⋅3=3

Cofactor matrix

𝐶=[245−4−123]
  • Entry (2, 3): remove row 2 and column 3

Minor:

[
]det([1210])=−2

Cofactor:

𝐶22=(−1)2+3⋅−2=−1⋅−2=2

Cofactor matrix

𝐶=[245−4−1232]
  • Entry (3, 1): remove row 3 and column 1

Minor:

[
]det([2345])=−2

Cofactor:

𝐶31=(−1)3+1⋅−2=1⋅−2=−2

Cofactor matrix

𝐶=[245−4−1232−2]
  • Entry (3, 2): remove row 3 and column 2

Minor:

[
]det([1305])=5

Cofactor:

𝐶32=(−1)3+2⋅−5=−1⋅5=−5

Cofactor matrix

𝐶=[245−4−1232−2−5]
  • Entry (3, 3): remove row 3 and column 3

Minor:

[
]det([1204])=4

Cofactor:

𝐶33=(−1)3+3⋅4=1⋅4=4

Cofactor matrix

𝐶=[245−4−1232−2−54]

Step 2. Cofactor matrix

𝐶=[245−4−1232−2−54]

Step 3. Transpose to get adjugate

adj(𝐴)=𝐶𝑇=[24−12−253−5−424]

Relation to inverse:

𝐴−1=1det(𝐴)adj(𝐴)ifdet(𝐴)≠0

Relation to identity:

𝐴⋅adj(𝐴)=adj(𝐴)⋅𝐴=det(𝐴)𝐼𝑛