M=MatrixForm[{{a,-b,-c,-d},{b,a,-d,c},{c,d,a,-b},{d,-c,+b,a}}]

Inverse[M]

MatrixForm[%38]

Inverse[(({{a, -b, -c, -d}, {b, a, -d, c}, {c, d, a, -b}, {d, -c, b, a}}))]

MatrixForm[%62]

%38.{1,0,0,0}

Inverse[(({{a, -b, -c, -d}, {b, a, -d, c}, {c, d, a, -b}, {d, -c, b, a}}))].{1,0,0,0}

{(((a^3)+a (b^2)+a (c^2)+a (d^2))/((a^4)+2 (a^2) (b^2)+(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)+(c^4)+2 (a^2) (d^2)+2 (b^2) (d^2)+2 (c^2) (d^2)+(d^4))),((-(a^2) b-(b^3)-b (c^2)-b (d^2))/((a^4)+2 (a^2) (b^2)+(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)+(c^4)+2 (a^2) (d^2)+2 (b^2) (d^2)+2 (c^2) (d^2)+(d^4))),((-(a^2) c-(b^2) c-(c^3)-c (d^2))/((a^4)+2 (a^2) (b^2)+(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)+(c^4)+2 (a^2) (d^2)+2 (b^2) (d^2)+2 (c^2) (d^2)+(d^4))),((-(a^2) d-(b^2) d-(c^2) d-(d^3))/((a^4)+2 (a^2) (b^2)+(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)+(c^4)+2 (a^2) (d^2)+2 (b^2) (d^2)+2 (c^2) (d^2)+(d^4)))}

Simplify[%41]

{(a/((a^2)+(b^2)+(c^2)+(d^2))),-(b/((a^2)+(b^2)+(c^2)+(d^2))),-(c/((a^2)+(b^2)+(c^2)+(d^2))),-(d/((a^2)+(b^2)+(c^2)+(d^2)))}

M2=MatrixForm[{{a,-b,-c,-d},{b,a,d,-c},{c,-d,a,b},{d,c,-b,a}}]

Inverse[M2]

Inverse[(({{a, -b, -c, -d}, {b, a, d, -c}, {c, -d, a, b}, {d, c, -b, a}}))]

%44.{1,0,0,0}

M.(%41)

Simplify[%45]

{(a/((a^2)+(b^2)+(c^2)+(d^2))),-(b/((a^2)+(b^2)+(c^2)+(d^2))),-(c/((a^2)+(b^2)+(c^2)+(d^2))),-(d/((a^2)+(b^2)+(c^2)+(d^2)))}