Standardized

matrix(a, b, c, d, e, f) = [a b c d e f]
translate(tx, ty=0)      = [1 0 0 1 tx ty] 
scale(sx, sy=sx)         = [sx 0 0 sy 0 0]
rotate(α)                = [cos(α) sin(α) -sin(α) cos(α) 0 0]
rotate(α, cx=0, cy=0)    = [1 0 0 1 cx cy] [cos(α) sin(α) -sin(α) cos(α) 0 0] [1 0 0 1 -cx -cy]
skewX(α)                 = [1 0 tan(α) 1 0 0]
skewY(α)                 = [1 tan(α) 0 1 0 0]

Proposed additions

mirror(a, b=-a)          = [a 0 0 b width*(1-a)/2 height*(1-b)/2]
flipX() = mirror(-1,1)   = [-1 0 0 1 width 0]
flipY() = mirror(1,-1)   = [1 0 0 -1 0 height]

mirror(a, b=1-a)         = [1-2*a 0 0 1-2*b width*a height*b]
flipX() = mirror(1,0)    = [-1 0 0 1 width 0]
flipY() = mirror(0,1)    = [1 0 0 -1 0 height]

Transformation matrix

T	1	2	3
1	a	c	e
2	b	d	f
3	0	0	1

Translation matrix

T	1	2	3
1	1	0	tx
2	0	1	ty
3	0	0	1

Scaling matrix

T	1	2	3
1	sx	0	0
2	0	sy	0
3	0	0	1

Rotation matrix

T	1	2	3
1	cos(α)	-sin(α)	0
2	sin(α)	cos(α)	0
3	0	0	1

Arbitrary anchor

T	1	2	3
1	cos(α)	-sin(α)	cx*(1-cos(α)) + cy*sin(α)
2	sin(α)	cos(α)	cy*(1-cos(α)) - cx*sin(α)
3	0	0	1

Horizontal skew matrix

T	1	2	3
1	1	tan(α)	0
2	0	1	0
3	0	0	1

Vertical skew matrix

T	1	2	3
1	1	0	0
2	tan(α)	1	0
3	0	0	1