Linear Algebra, 2020a

366 Chapter Four. Determinants
Note that the order of the subscripts in the matrix of cofactors is opposite to
the order of subscripts in the other matrix; e.g., along the first row of the matrix
of cofactors the subscripts are 1, 1 then 2, 1, etc.
1.8 Definition The matrix adjoint (or the classical adjoint or adjugate) tothe
square matrix T is
⎛
⎞
T 1,1 T 2,1 ... T n,1
T 1,2 T 2,2 ... T n,2
adj(T) =
⎜
⎝
⎟
.
⎠
T 1,n T 2,n ... T n,n
where the row i, column j entry, T j,i , is the j, i cofactor.
1.9 Theorem Where T is a square matrix, T · adj(T) =adj(T) · T = |T| · I. Thusif
T has an inverse, if |T| ≠ 0, then T −1 =(1/|T|) · adj(T).
Proof Equations (∗) and (∗∗).
1.10 Example If
then adj(T) is
⎛
1 −1
⎛
⎜
T 1,1 T 2,1 T ⎞
∣0 1 ∣
3,1
⎟
2 −1
⎝T 1,2 T 2,2 T 3,2 ⎠=
−
1 1 ∣
T 1,3 T 2,3 T 3,3 ⎜
⎝
2 1
∣1 0∣
⎛
⎜
1 0 4
⎞
⎟
T = ⎝2 1 −1⎠
1 0 1
−
0 4
∣0 1∣
1 4
∣1 1∣
−
1 0
∣1 0∣
QED
0 4
⎞
∣1 −1∣
⎛
⎞
1 4
1 0 −4
⎜
⎟
−
= ⎝−3 −3 9 ⎠
2 −1∣
1 0
⎟
−1 0 1
⎠
∣2 1∣
and taking the product with T gives the diagonal matrix |T| · I.
⎛
⎜
1 0 4
⎞ ⎛
⎞ ⎛
⎞
1 0 −4 −3 0 0
⎟ ⎜
⎟ ⎜
⎟
⎝2 1 −1⎠
⎝−3 −3 9 ⎠ = ⎝ 0 −3 0 ⎠
1 0 1 −1 0 1 0 0 −3
The inverse of T is (1/ − 3) · adj(T).
⎛
⎞
1/−3 0/−3 −4/−3
T −1 ⎜
= ⎝−3/−3 −3/−3 9/−3
−1/−3 0/−3 1/−3
⎟
⎠ =
⎛
⎞
−1/3 0 4/3
⎜
⎟
⎝ 1 1 −3 ⎠
1/3 0 −1/3