Given that A = [[4, 9, 2], [-8, 6, -3], [3, 1, -5]], Evaluate |A| + |A^T|. Hence or otherwise solve the equations 4x + 9y + 2z = 21, -8x + 6y - 3z = 41, 3x + y - 5z = -73.

Here are the solutions to your questions. Q1. Given that A = 4 & 9 & 2 \\ -8 & 6 & -3 \\ 3 & 1 & -5 . Evaluate |A| + |A^T| Step 1: Calculate the determinant of matrix A. |A| = 4 6 & -3 \\ 1 & -5 - 9 -8 & -3 \\ 3 & -5 + 2 -8 & 6 \\ 3 & 1 |A| = 4((6)(-5) - (-3)(1)) - 9((-8)(-5) - (-3)(3)) + 2((-8)(1) - (6)(3)) |A| = 4(-30 + 3) - 9(40 + 9) + 2(-8 - 18) |A| = 4(-27) - 9(49) + 2(-26) |A| = -108 - 441 - 52 |A| = -601 Step 2: Use the property that |A| = |A^T|. Therefore, |A| + |A^T| = |A| + |A| = 2|A|. |A| + |A^T| = 2(-601) = -1202 The value of |A| + |A^T| is -1202. Solve the equations: 4x + 9y + 2z = 21 -8x + 6y - 3z = 41 3x + y - 5z = -73 Step 3: Recognize that the coefficient matrix of the system is A. We use Cramer's Rule to solve for x, y, z. We already found |A| = -601. Calculate the determinant of A_x (matrix A with the first column replaced by the constant terms): A_x = 21 & 9 & 2 \\ 41 & 6 & -3 \\ -73 & 1 & -5 |A_x| = 21((6)(-5) - (-3)(1)) - 9((41)(-5) - (-3)(-73)) + 2((41)(1) - (6)(-73)) |A_x| = 21(-30 + 3) - 9(-205 - 219) + 2(41 + 438) |A_x| = 21(-27) - 9(-424) + 2(479) |A_x| = -567 + 3816 + 958 = 4207 x = (|A_x|)/(|A|) = (4207)/(-601) = -7 Step 4: Calculate the determinant of A_y (matrix A with the second column replaced by the constant terms): A_y = 4 & 21 & 2 \\ -8 & 41 & -3 \\ 3 & -73 & -5 |A_y| = 4((41)(-5) - (-3)(-73)) - 21((-8)(-5) - (-3)(3)) + 2((-8)(-73) - (41)(3)) |A_y| = 4(-205 - 219) - 21(40 + 9) + 2(584 - 123) |A_y| = 4(-424) - 21(49) + 2(461) |A_y| = -1696 - 1029 + 922 = -1803 y = (|A_y|)/(|A|) = (-1803)/(-601) = 3 Step 5: Calculate the determinant of A_z (matrix A with the third column replaced by the constant terms): A_z = 4 & 9 & 21 \\ -8 & 6 & 41 \\ 3 & 1 & -73 |A_z| = 4((6)(-73) - (41)(1)) - 9((-8)(-73) - (41)(3)) + 21((-8)(1) - (6)(3)) |A_z| = 4(-438 - 41) - 9(584 - 123) + 21(-8 - 18) |A_z| = 4(-479) - 9(461) + 21(-26) |A_z| = -1916 - 4149 - 546 = -6611 z = (|A_z|)/(|A|) = (-6611)/(-601) = 11 The solution to the system of equations is x = -7, y = 3, z = 11. Q2.(a) Evaluate in polar form (i) (16 75^)/(2 15^) Step 1: To divide complex numbers in polar form, divide their magnitudes and subtract their angles. (16 75^)/(2 15^) = ((16)/(2)) (75^ - 15^) = 8 60^ The result is 8 60^. (ii) (10 )/(4) × 12 ()/(2)6 -()/(3) Step 1: First, multiply the complex numbers in the numerator. Multiply magnitudes and add angles. 10 ()/(4) × 12 ()/(2) = (10 × 12) (()/(4) + ()/(2)) = 120 (()/(4) + (2)/(4)) = 120 (3)/(4) Step 2: Now, divide the result from Step 1 by the denominator. Divide magnitudes and subtract angles. (120 3)/(4)6 -()/(3) = ((120)/(6)) ((3)/(4) - (-()/(3))) = 20 ((3)/(4) + ()/(3)) = 20 ((9)/(12) + (4)/(12)) = 20 (13)/(12) The result is 20 (13)/(12). Q2.(b) If Z = R + j L + (1)/(j C), express Z in (a+jb) form when R=10, L=5, C=0.04 and =4. Step 1: Substitute the given values into the expression for Z. Z = 10 + j(4)(5) + (1)/(j(4)(0.04)) Step 2: Simplify the terms. Z = 10 + j20 + (1)/(j(0.16)) Step 3: Simplify the fraction (1)/(j(0.16)) by multiplying the numerator and denominator by j (or -j). (1)/(j(0.16)) = (1)/(j(0.16)) × (j)/(j) = (j)/(j^2(0.16)) Since j^2 = -1: = (j)/(-(0.16)) = -(1)/(0.16)j Convert the fraction to a decimal: (1)/(0.16) = (100)/(16) = (25)/(4) = 6.25. So, (1)/(j(0.16)) = -6.25j. Step 4: Substitute this back into the expression for Z and combine the imaginary terms. Z = 10 + j20 - j6.25 Z = 10 + j(20 - 6.25) Z = 10 + j13.75 The expression for Z in (a+jb) form is 10 + j13.75. Send me the next one 📸