Find the determinant of the matrix M 3 2x3 4 2x2 4x3 0

Find the determinant of the matrix M = [3 - 2x^3 - 4 + 2x^2 + 4x^3 0 -x^3 1 + x^2 + 2x^3 0 10 + 4x^2 - 20 -8x^2 -2 -2x^2] and use the adjoint method to find M^-1. det(M) =, M^-1 = [].

Solution

Det (M) = (3-2x²)[ (1+x²+2x³)(-2-2x²)]- (-4+2x²+4x³)[-x³(-2-2x²)] = [(2x²-3)(2x²+2)(2x³+x²+1)]– [(4x³+2x²-4) x³(2x+2)] = (8x⁷+4x⁶-4x⁵+2x⁴-12x³-8x²-6)-(8x⁷+12x⁶+4x⁵-8x⁴-8x³) = 16x⁶-8x⁵ +10x⁴ -4x³-8x²-6.

The Minors are computed as under:

Then,, the co- factor matrix is

-(4x⁵+2x⁴+4x³+4x²+2)

-(2x⁵+2x³)

-(4x⁴ +14x²+10)

(8x⁵ +4x⁴+8x³-4x²-8)

(4x⁵ +4x³-3x²-6)

(8x⁴ +40x²+20)

0

0

-(3x²+3)

The adjoint matrix of M is Adj(M) = P (say) =

-(4x⁵+2x⁴+4x³+4x²+2)

(8x⁵ +4x⁴+8x³-4x²-8)

0

-(2x⁵+2x³)

(4x⁵ +4x³-3x²-6)

0

-(4x⁵ +4x³-3x²-6)

(8x⁴ +40x²+20)

-(3x²+3)

Then M^-1 = P/det(M)

-(4x5+2x4+4x3+4x2+2)	-(2x5+2x3)	-(4x4 +14x2+10)
(8x5 +4x4+8x3-4x2-8)	(4x5 +4x3-3x2-6)	(8x4 +40x2+20)
0	0	-(3x2+3)