# Essays/Minors

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A minor of size one less than the size of a matrix obtains by removing row i and column j ; the array of such minors can be computed using the outfix (\.):

```   ] m=: i.3 4
0 1  2  3
4 5  6  7
8 9 10 11

minors=: 1&(|:\.)"2^:2

\$ minors m
3 4 2 3
<"_2 minors m
┌───────┬───────┬──────┬──────┐
│5  6  7│4  6  7│4 5  7│4 5  6│
│9 10 11│8 10 11│8 9 11│8 9 10│
├───────┼───────┼──────┼──────┤
│1  2  3│0  2  3│0 1  3│0 1  2│
│9 10 11│8 10 11│8 9 11│8 9 10│
├───────┼───────┼──────┼──────┤
│1 2 3  │0 2 3  │0 1 3 │0 1 2 │
│5 6 7  │4 6 7  │4 5 7 │4 5 6 │
└───────┴───────┴──────┴──────┘
```

Complementary indexing also lends itself to the computation of minors:

```   ] i=: <&.>&.> {i.&.>\$m
┌─────────┬─────────┬─────────┬─────────┐
│┌───┬───┐│┌───┬───┐│┌───┬───┐│┌───┬───┐│
││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐││
│││0│││0│││││0│││1│││││0│││2│││││0│││3│││
││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘││
│└───┴───┘│└───┴───┘│└───┴───┘│└───┴───┘│
├─────────┼─────────┼─────────┼─────────┤
│┌───┬───┐│┌───┬───┐│┌───┬───┐│┌───┬───┐│
││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐││
│││1│││0│││││1│││1│││││1│││2│││││1│││3│││
││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘││
│└───┴───┘│└───┴───┘│└───┴───┘│└───┴───┘│
├─────────┼─────────┼─────────┼─────────┤
│┌───┬───┐│┌───┬───┐│┌───┬───┐│┌───┬───┐│
││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐│││┌─┐│┌─┐││
│││2│││0│││││2│││1│││││2│││2│││││2│││3│││
││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘│││└─┘│└─┘││
│└───┴───┘│└───┴───┘│└───┴───┘│└───┴───┘│
└─────────┴─────────┴─────────┴─────────┘
\$ i{m
3 4 2 3
<"2 i{m
┌───────┬───────┬──────┬──────┐
│5  6  7│4  6  7│4 5  7│4 5  6│
│9 10 11│8 10 11│8 9 11│8 9 10│
├───────┼───────┼──────┼──────┤
│1  2  3│0  2  3│0 1  3│0 1  2│
│9 10 11│8 10 11│8 9 11│8 9 10│
├───────┼───────┼──────┼──────┤
│1 2 3  │0 2 3  │0 1 3 │0 1 2 │
│5 6 7  │4 6 7  │4 5 7 │4 5 6 │
└───────┴───────┴──────┴──────┘

minors1=: <&.>&.>@{@(i.&.>"_)@\$ { ]

(minors -: minors1) m
1
```

Laplace expansion of the determinant of a square matrix:

```   det=: -/ .*

] m=: _8 + 5 5 ?@\$ 20
2 _1 10  1 _6
4  7 _3  7  5
10  2 _8 _3  1
_2  6 _8 _2 _4
1  0  7  9 _4

] cofactors=: (_1^+/&>{i.&.>\$m) * det minors m
_2560 _8740 _7780  6540   460
_640 _8510 _3870   810 _5110
_5760  3490  3010  _630  2410
2240 _4425  2215   355  5235
_640  7770  5370 _9310  5890

det m
_70400

i=: ?#m
+/ (i{m) * i{cofactors
_70400

j=: ?{:\$m
+/ (j{"1 m) * j{"1 cofactors
_70400

+/ m * cofactors
_70400 _70400 _70400 _70400 _70400

+/"1 m * cofactors
_70400 _70400 _70400 _70400 _70400
```

See also

Contributed by Roger Hui. The two phrases for minors appear in the Special Matrices section of J Phrases; the first phrase can be found on the green J mug.