3x1 - 2x2 + x3 = 4 x1 + x2 - x3 = 2 x1 + x3 = 1 2. The given...

3x_1 - 2x_2 + x_3 = 4 x_1 + x_2 - x_3 = 2 x_1 + x_3 = 1 2. The given linear equation system Ax = b above, please use the augmented matrix [A|I] to find A?¹. a) Find adj(A) in Ax = b. b) Use Cramer's rule to solve the system. c) Let v1 v2 v3 be vectors R³ and A = [v1,v2,v3]. Determine whether {v1,v2,v3} are linearly dependent or linearly independent in R³.

Transcript

00:01 So you were given a system of equations in three variables, and the first thing we want to do is create the augmented matrix.

00:08 And to create the augmented matrix, we just take the coefficients of each of the equations.

00:24 And then we want to use that augmented matrix to solve the equation.

00:29 So legal moves or moves that you're allowed to do would be to add any two rows and replace one of the rows with its sum.

00:39 You can multiply any row by a non -zero number, or you can do a combination of the two.

00:45 Multiply any row by a non -zero number, add it to another row, and replace one of those rows.

00:51 So the first thing i'm going to do, to go from my first matrix to my second, is i'm going to multiply negative 1 times row 2, add it to row 1, and i'm going to replace row 2.

01:07 So if i multiply negative 1 times row 2, i'm going to get a negative 2, a positive 3, a positive 3, and a negative 22.

01:19 And then i'm going to add those numbers to row 1, and i'm going to put the answer in row 2.

01:26 So that means row 1 and row 3 are not changing.

01:31 So now when i add the new row two with row one, i get 0, 6, 4, and negative 12.

01:49 So my next step is i want to work on cleaning up row 3 a little bit.

01:55 So this time i'm going to take negative 2 times row 1, and i'm going to add it to row 3, and i'm going to put it in place of row 3.

02:05 So that means row 1 is not changing and row 2 is not changing.

02:18 But when i multiply row 2, or sorry, row 1 by a negative 2, i get negative 4, negative 6, negative 2, and negative 20.

02:29 And when i add that new set of numbers with the bottom row, i then get 0, negative 8, 1, and negative 20.

02:41 So to create my next matrix, i'm just going to take row two and multiply it by a half.

02:59 And i'm going to put it into row two.

03:03 So that means row one remains unchanged, row three remains unchanged.

03:13 And the new row two would be 0, 3, 2, negative 6.

03:23 For my next transformation, i'm going to take.

03:31 Row 2 multiply by negative 1 add it to row 1 and replace row 1 so that means row 2 is unchanged row 3 is unchanged row 2 is going to change to a 0 negative 3 negative 2 positive 6 so when i add the new row 2 to row 1 i get 0 try again i get 2, 0, negative 1, 16.

04:23 For my next one, i'm going to take row 2, and i'm going to multiply it by 8, and i'm going to take row 3 and multiply it by 3, and i'm going to replace row 3.

04:48 So that means row 1 and 2 remain the same.

04:51 When i multiply row 2 by 8, i get 0, 24, 16, and negative 48.

05:09 And when i multiply row 3 by 3, i get 0, negative 24, 3, and negative 66.

05:22 And when i add those two sets of numbers, i get 0, 0 ,000, 3, and negative 66...

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