Question

HW10P1 (12 points) For the following system of equations 3x1 - x2 + 4x3 = -9 -4x, + x2 + 2xy = -4 2x1 + x3 = 0 a. (2 pts) Write the linear system in the format, A x = b. b. (2 pts) Find the determinant of the matrix A by using an expansion along row 1. c. (2 pts) Find the determinant of the matrix A by using an expansion along column 2. d. (3 pts) Perform the row operation R1 + R2 R2 and find the determinant of the new matrix by expansion on column 2. Compare the new determinant with that of matrix A obtained in band c. Show that your result agrees with the determinant properties e. (2 pts) Interchange column 1 with column 2 of matrix A, and find the determinant of the transformed matrix with the expansion along row 1 of the new matrix. Compare the new determinant with that of matrix A obtained in b and c. Show that your result agrees with the determinant properties. f. (1 pt) Based on your results of parts b, c, d, and e, explain why matrix A is invertible. Do NOT determine the inverse.
(Has to be done by hand)

Answer 1

using Cramer's rule, (HW10 P2) To use cramer's rule to solve a system of three equations with three unknowns, we need to follow these steps: Step 1: Find the determinant, D, by using the ny, and a values from the problem. Pooblem. Step 2: Find the determinant , Dx, by replacing the H-values in the first column with the values after the equal sign leaving the y and 2 columns unchanged. Step 3: Find the determinant, Dy, by replacing the y-values in the second column with the values after the equal sign leaving the x and 2 columns unchanged. Step 4: Find the determinant ; Dz g by replacing the 2-values in the third column with the values after the equal sign leaving the y and a columns unchanged. Step 5: once we have calculated the values Do Dug Dy and Da We can apply cramer's rule to find x,y and a.

use cramer's rule to solve the given equation: 3x – 12 +413 = -9 uni + H2 + 2x3 =-4 2x +13=0 Step 1: Find the determinant, D, by using the x,y and a values from the problem. = 3 - 7 1-97 -4 I 2 = 1-4 [2 o i = 3(1-0)+1(-4-4) 44 (0-2) - 3-8-8. - 13 Step 2: Find the determinant , Dx, by replacing the n-values in the first column with the values after the equal sign leaving the y and a columns unchanged. 1-9 -1 4 - -4 , 2 OO! + - + +J - x=-631--01?+4161 2-9(1-0) +1(-4-1) + 460-0) --9-4+0 -13

Step 3. Find the determinant, Dy, by replacing the y-values in the second column with the values after the equal sigh leaving the x and Z columus unchanged. Dy = 13 e t 1-4 –4 2 =317?/--0) 271+4177 = 3(-4-0) +91-4-4) +4 (0+8) = -12-72432 - 84 +32 - 52 Step 4: Find the determinant, Dz, by replacing the z-values in the third columns with the values after the equal sign leaving the n and y columns unchanged. Da= 13 i ta -4 Iz I o - 4 o -36-77-6-00346-0036 -3(0+0) + 1 (0+8) -9 (0-2) = 0+8+18 5 26

values of Step 5: Use cramer's rule to find the x,y and z. Generally, the answer is written as on order triple (1, 4,-2) representing the lo y and a values.