Wolfram Alpha:

```Determinants
------------

D = | a b c |
| d e f |
| g h i |

D = a(ei - hf) - b(di - fg) + c(dh - eg]

or

= -d(bi - ch) + e(ai - cg] - f(ah - bg]

or

= g(bf - ce) - h(af - cd] + i(ae - bd]

Example:

D = | 0 1 2 |
| 3 4 5 |
| 6 7 0 |

D = 0(4*0 - 5*7) - 1(3*0 - 5*6) + 2(3*7 - 4*6)
= 24

or

D = -3(1*0 - 2*7) + 4(0*0 - 2*6) - 5(0*7 - 1*6)
= 24

Matrices
--------

-     -
M = | a b c |
| d e f |
| g h i |
-     -

Transpose:

-     -
MT = | a d g |
| b e h |
| c f i |
-     -

Cofactor Matrix:

A, B, C, D, E, F, G and I are determinants

A = | e f |  B = | d f |   C = | d e |
| h i |      | g i |       | g h |

D = | b c |  E = | a c |   F = | a b |
| h i |      | g i |       | g h |

G = | b c |  H = | a c |   I = | a b |
| e f |      | d f |       | d e |

-        -
C(M) = |  A -B  C |
| -D  E -F |
|  G -H  I |
-        -

-     -
adj(M) = C(M)T = | A D G |
| B E H |
| C F I |
-     -

Identity:

-          -
I = | 1 0 0 0 .. |
| 0 1 0 0 .. |
| 0 0 1 0 .. |
| 0 0 0 1 .. |
| ........ 1 |
-          -

Inverse:

M-1 = (1/det M)adj(M)  implies MM-1 = I

Trace:

Tr(A) = a + e + i

Multiplication:

-     -
A = | a b c |
| d e f |
-     -
-   -
B = | h i |
| j k |
| l m |
-   -
-   -
A.B = | W X |
| Y Z |
-   -

W = (a, b, c). (h, j, l) = ah + bj + cl
X = (a, b, c). (i, k, m) = ai + bk + cm
Y = (d, e, f). (h, j, l) = dh + ej + fl
Z = (d, e, f). (i, k, m) = di + ek + fm

Differentiation:

Let y = f(x)

-                          -
∂y/∂x  = | ∂y1/∂x1 ∂y1/∂x2 ... ∂y1/∂xm |
| ∂y2/∂x1 ∂y2/∂x2 ... ∂y2/∂xm |
| .....................       |
| ∂yn/∂x1 ∂y2/∂x2 ... ∂yn/∂xm |
-                          -

Examples:

M:
-       -
| 1+i  2i |
| -i   1  |
-       -

MT:
-      -
| 1+i -i |
| 2i   1 |
-      -

M*:
-       -
|  1   i  |
| -2i i+1 |
-       -

-      -
| 1 -2i  |
| i  i+1 |
-      -

M-1:
-     -
(-1-i)/2| 1 -2i |
| i 1+i |
-     -

Tr(M): 2 + i

Hermitian Matrix
----------------

Definition:

- M† = M where † means conjugate transpose (M*)T
- Complex entries but main diagonal must be real
(including 0).
- Must be square.

Examples:

M:
-           -
|   1  1-i  2 |
|  1+i  3   i |
|   2  -i   0 |
-           -

M*:
-            -
|   1   1+i  2 |
|  1-i   3   1 |
|   2    i   0 |
-            -

MT:
-          -
|   1  1-i  2 |
|  1+i  3   1 | = M
|   2  -i   0 |
-           -

Skew Hermitian (aka anti-Hermitian) Matrix
------------------------------------------

Definition:

- M† = -M where † means conjugate transpose (M*)T
- Complex entries but main diagonal must be imaginary
(including 0).
- Must be square.

Examples:

M:
-          -
|   i  1-i  2 |
| -1-i 3i   i |
|  -2   i   0 |
-          -

M*:
-            -
|  -i  1+i   2 |
| -1+i -3i  -i |
|  -2  -i    0 |
-            -

MT:
-           -
| -i  -1+i  -2 |
| 1+i  -3i  -i | = -M
|  2   -i    0 |
-            -

Orthogonal Matrix
-----------------

Definition:

- MT = M-1
- MTM = MMT = I
- Must be square.
- Must be real.
- |det M| = 1 (the inverse is not necessarily true,
a matrix can have a |det| = 1 and not be orthogonal)

Examples:

M:
-
| cosθ -sinθ |
| sinθ  cosθ |
-           -

|det M| = 1

MT:
-
|  cosθ  sinθ |
| -sinθ  cosθ |
-           -

M-1:
-           -
|  cosθ  sinθ |
| -sinθ  cosθ |
-           -

Unitary Matrix
--------------

Complex analog of an orthogonal matrix.

U† = U-1 where † means conjugate transpose.
Thus U†U = I

Unitary matrices have complex determinants.
with |det U| = 1.

Special Unitary Matrix
----------------------

U† = U-1 where † means conjugate transpose.
Thus U†U = I

Special unitary matrices must have det U = +1
(not modulus 1)

```