Prove that `|[1+a,1,1,1],[1,1+a,1,1],[1,1,1+a,1],[1,1,1,1+a]|=a^(3)(a+4)`

Prove that abs([1,1,1],[1,1+a,1],[1,1,1+b]) = ab

Prove that : Det[[1+a,1,1],[1,1+b,1],[1,1,1+c]]=abc(1+1/a+1/b+1/c)

Prove: |(1+a,1, 1),( 1, 1+a, 1),(1, 1, 1+a)|=a^3+3a^2

If A=[[1,1,1],[1,2,1],[1,1,2]] ,then prove that A^(-1)=[[3,-1,-1],[-1,1,0],[-1,0,1]]

If A = [(2,2,1), (1,3,1), (1,2,2)] then A^-1+(A-5I) (AI)^2 = (i) 1/ 5 [[4,2, -1], [-1,3,1], [-1,2,4]] (ii) 1/5 [[4, -2, -1], [-1, 3, -1], [-1, -2,4]] (iii) 1/3 [[4,2, -1], [-1,3,1], [-1,2,4]] (iv) 1/3 [[4, -2, -1], [-1,3, -1], [-1, -2,4]]

Prove that: 1 + tan^-1 (1) + tan^-1 (3) = π

Using properties of determinants prove that ((1,1,1+3x),(1+3y,1,1),(1,1+3z,1)=9(3xyz+xy+yz+zx)

Prove that : tan^(-1)(1/2) + tan^(-1)(1/3) = tan^(-1)(3/5) + tan^(-1)(1/4) = pi/4

Prove that tan ^(-1)(1/5) + tan^(-1)(1/7) +tan^(-1)(1/3)+ tan ^(-1)(1/8) = pi/4

Prove that tan^(-1) (1/8) +tan^(-1) (1/5) =tan^(-1) (1/3)