Question

Solve the following systems of equation using the formula by Cramer and with the inverse matrices.

4x₁ - x₂ + 2x₃ = 8

-x₁ + 2x₂ = -7

x₁ - 3x₂ - 5x₃ = 2

Answer 1

system Ax=b is

find determinant of A

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replace first column with vector b

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replace second column with vector b

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replace third column with vector b

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here matrix A is

4	-1	2
-1	2	0
1	-3	-5

add the Identity Matrix to the right of our matrix

4	-1	2	1	0	0
-1	2	0	0	1	0
1	-3	-5	0	0	1

by Gauss-Jordan Elimination

Divide row1 by 4

1	-1/4	1/2	1/4	0	0
-1	2	0	0	1	0
1	-3	-5	0	0	1

Add (1 * row1) to row2

1	-1/4	1/2	1/4	0	0
0	7/4	1/2	1/4	1	0
1	-3	-5	0	0	1

Add (-1 * row1) to row3

1	-1/4	1/2	1/4	0	0
0	7/4	1/2	1/4	1	0
0	-11/4	-11/2	-1/4	0	1

Divide row2 by 7/4

1	-1/4	1/2	1/4	0	0
0	1	2/7	1/7	4/7	0
0	-11/4	-11/2	-1/4	0	1

Add (11/4 * row2) to row3

1	-1/4	1/2	1/4	0	0
0	1	2/7	1/7	4/7	0
0	0	-33/7	1/7	11/7	1

Divide row3 by -33/7

1	-1/4	1/2	1/4	0	0
0	1	2/7	1/7	4/7	0
0	0	1	-1/33	-1/3	-7/33

Add (-2/7 * row3) to row2

1	-1/4	1/2	1/4	0	0
0	1	0	5/33	2/3	2/33
0	0	1	-1/33	-1/3	-7/33

Add (-1/2 * row3) to row1

1	-1/4	0	35/132	1/6	7/66
0	1	0	5/33	2/3	2/33
0	0	1	-1/33	-1/3	-7/33

Add (1/4 * row2) to row1

1	0	0	10/33	1/3	4/33
0	1	0	5/33	2/3	2/33
0	0	1	-1/33	-1/3	-7/33

inverse matrix:

10/33	1/3	4/33
5/33	2/3	2/33
-1/33	-1/3	-7/33

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for system