Let a and b be 3 ã 3 matrices with det(a) = 4 and det(b) = 5. find the value of (a) det(ab)

(a) |AB| = |A| |B| = -1 * 2 = -2. 

(b) |B^5| = |B|^5 = 2^5 = 32. 

(c) Since A is 4 x 4, 
|2A| = 2^4 * |A| = 16 * -1 = -16. 

(d) Since determinants are invariant under transposition, 
|A^t A| = |A^t| |A| = |A| |A| = (-1)(-1) = 1. 

(e) |B^(-1) AB| = |B^(-1)| |A| |B| 
= (1/|B|) |A| |B| 
= |A| 
= -1. 

I hope this helps!