-4 6 6 5 0 Let A -17 4 4 B 1 3 1 -6 0 -6 0 0 DC- 8 0 0 8 0 1 10 0 0 4 AB BC Prove WITHOUT using the product property of determinants: If M, N are upper triangular matrices, |MN| = |M|N|. This question will be graded after the due date.
0 7. Let A= 71 -2 1 2 -1 -1 (1 2 2 , B= 1 -2 1 -1 10 and C= 1 -3 2 100 1 100 (a) Compute det A, det B, det C, det(AB) and det(A + B). (b) Verify that A and C are inverse matrices and use this fact to (i) solve the simultaneous equations: x – 2y + z = 2 2x – y - 2= -1 x + 2y + 2z = 3...
4. Let 13 1 1) A= 1 1 1 (105) (a) Use the method of augmented matrices to compute the inverse of A. (b) Use the method of the inverse to solve the system of equations 3x + y + z = 5 2+y+z = 3 I + 5z = 6 ]AO PIO @ 91 * A
Consider the following three 2x2 matrices (Pauli's matrices): ?x=(0 1) ?y=(0 ?i) ?z=(1 0 ) 1 0 i 0 0 ?1 4. Show that Pauli's matrices are Hermitian. 5. Compute the column vector corresponding to ?x|b? where |b? =1 i 6. Compute the expectation values of ?x in state |b? : ?xb=?b|?x|b? ______ ?b|b?
13 please 8. b. -2 3 0 0 0 0 -1 2 0 0-4 0 3 0-2 0 3 0 0 -2 0 3 0 4 o0-1 6 0 0 1 o 2 6 0 0 -1 6 10. For any positive integer k, prove that det(4t) - de(A)*. 11. Prove that if A is invertible, then den(A-1)- I/der(A) - det(4)- 12. We know in general that A-B丰B-A for two n x n matrices. However, prove that: det(A . B)-det(B...
4. Let 3 f(x, y, z) = x’yz-xyz3, 4 P(2, -1, 1), u =< 0, > 5 a). Find the gradient of f. b). Evaluate the gradient at the point P. c). Find the rate of change of f at the point of P in the direction of the vector u.
Let X be a Poisson (mean = 5) and Let Y be a Poisson (mean = 4). Let Z = X + Y. Find P( X = 3 | Z = 6). Assume X and Y are independent. Show answers for P(A), P(B), P(AB), and and hence P(A|B). Here A = [X = 3], B = [Z = 6]
Let X be a random variable with CDF z<0 G()=/2 0 <IS2 z>2 1 Suppose Y = X2 is another random variable, find (a) P(1/2 X 3/2), (b) P(1s X< 2) (c) P(Y X) (d) P(X 2Y). (f) If Z VX, find the CDF of Z. (d) P(X+Y 3/4)
1. Evaluate the following determinants: 8 7 3 4 0 2 (a) 4 0 1 (c) 6 0 3 6 03 8 2 3 ab (e) b c с a b 1 1 1 2 3 (b) 4 7 5 3 6 9 4 -2 (d) 8 11 4 х (f) 3 9 5 0 y 2 -18 7
2 seperate questions the last picture is part of the second question ( multiple choice) Let A and B benxn invertible matrices, then det(B-1 AB) = 0 det(A) det(B) -det(A) (1 2-3 51 The augmented matrix for a system is given as, 0 1 4-6. Find the general solution or state that 0 0 0 0 there is no solution x=5-2y+32 y=-6-42 z is free x=17 y=-6 z=1 0 O X 17 [11] y -6 +1 -4. N 0