# Determinant of 2-forms

I have matrix 4x4 and elements of the matrix are 2-forms. How to calculate determinant (in Mathematica 11) if this matrix using external product instead of normal product? I use components of Riemann tensor of forth rank RUddd and calculate 2-form of curvature

rt[a_, b_] :=
Sum[FullSimplify[RUddd][[a, k, b, l]] TensorWedge[x[k], x[l]], {k, 1,
4}, {l, 1, 4}]


Then make matrix

Mrt = Table[
rt[a, b], {a, 1, 4}, {b, 1, 4}] /. {x[1]\[TensorWedge]x[1] -> 0,
x[2]\[TensorWedge]x[2] -> 0, x[3]\[TensorWedge]x[3] -> 0,
x[4]\[TensorWedge]x[4] -> 0};


and then obtain matrix which I need

gg = (DiagonalMatrix[{1, 1, 1, 1}] + I/(2 \[Pi]) Mrt);


Now I need to calculate determinant of this matrix but using wedge product instead of normal product.

• Is this about the software Mathematica by Wolfram? Commented Jan 11, 2017 at 15:52
• Yes, I'd like to solve by using Mathematica 11
– nail
Commented Jan 11, 2017 at 16:35
• Then please post the code you already have for the matrix (elements). Commented Jan 11, 2017 at 16:37
• @ Marius Ladegård Meyer I added code. Because product of 2-forms is commutative I calculated determinant with normal product and after calculation I changed by hand product of 2-forms for wedge product. But I think it possible to calculate by using power of Mathematica.
– nail
Commented Jan 11, 2017 at 16:50
• Can you, please, clarify two items for me. First, how should x[1] be interpreted? Is it your symbol for the 1-form basis? Or, is it a numeric array? I'm thinking that is all it could be, to be used with TensorWedge[]. Second, the gg[[1,1]] component looks to me like a scalar added to the sum of wedge products of the 1-form basis. For me it looks like a scalar added to a 2-form basis. When the determinant of gg is expanded, it looks like there would be scalars added to 2-forms added to 4-forms, etc. Commented Jan 13, 2017 at 10:15