This is how we enter a matrix in Mathematica:

{{1,-1,1,-1},{0,2,-8,18},{0,0,8,-48},{0,0,0,32}}

Now find its inverse:

Inverse[{{1,-1,1,-1},{0,2,-8,18},{0,0,8,-48},{0,0,0,32}}]

Since I use LATEX, this is how I export a matrix as LATEX code.

TeXForm[{{1,(1/2),(3/8),(5/16)},{0,(1/2),(1/2),(15/32)},{0,0,(1/8),(3/16)},{0,0,0,(1/32)}}]

Now verify whether the inverse is correct:

{{1,(1/2),(3/8),(5/16)},{0,(1/2),(1/2),(15/32)},{0,0,(1/8),(3/16)},{0,0,0,(1/32)}}.{{1,-1,1,-1},{0,2,-8,18},{0,0,8,-48},{0,0,0,32}}

Now show the inverse in traditional matrix notation:

MatrixForm[{{1,(1/2),(3/8),(5/16)},{0,(1/2),(1/2),(15/32)},{0,0,(1/8),(3/16)},{0,0,0,(1/32)}}]

Now find the transpose of this matrix. I need the last row of the transpose

Transpose[{{1,(1/2),(3/8),(5/16)},{0,(1/2),(1/2),(15/32)},{0,0,(1/8),(3/16)},{0,0,0,(1/32)}}]

Now create a linear combination of the multiple angle trig functions with the given coefficients

{(5/16),(15/32),(3/16),(1/32)}.{1,Cos[2 t],Cos[4 t],Cos[6 t]}

I want to copy this to the website as LATEX code

TeXForm[(5/16)+(15/32) Cos[2 t]+(3/16) Cos[4 t]+(1/32) Cos[6 t]]

On the website I explained why the following identity is true:

FullSimplify[(5/16)+(15/32) Cos[2 t]+(3/16) Cos[4 t]+(1/32) Cos[6 t]]