Question

A=l

−2 −7 −45

4 5 27
1 3 20

.
Find the third column of
A−1 ( subscript -1)
without computing the other two columns.
How can the third column of
A−1 ( subscript -1)
be found without computing the other​ columns?
A.
Row reduce the augmented matrix​ [AI3​( not 13 but capital I subscript 3].
B.
Row reduce the augmented matrix​ [A e3​],where e3 is the third column of I3.( capital I subscript 3)
C.
Row reduce the augmented matrix

A

e3

​,
where e3
is the third row of I3.( capital I subscript 3- not 13)
D.
Solve the equation
Ae3=b
for
e3​,
where
e3
is the third column of I3( capital I subscript 3, not 13)
and b is the third column of
A−1( A subscript -1 not A subtract 1).
The third column of
A−1 ( A subscript -1)
is __________________

​(Type an integer or decimal for each matrix​ element.)

Answer 1

In order to compute the third column of A^-1, we have to solve the equation AX = (0 0 1)^T, where (0,0,1)^T is the third column of I₃. Let B = [A|(0,0,1)^T]=

-2	-7	-45	0
4	5	27	0
1	3	20	1

To solve equation AX = (0 0 1)^T , we can row reduce B as under:

Multiply the 1st row by -1/2

Add -4 times the 1st row to the 2nd row

Add -1 times the 1st row to the 3rd row               

Multiply the 2nd row by -1/9

Add 1/2 times the 2nd row to the 3rd row

Add -7 times the 3rd row to the 2nd row

Add -45/2 times the 3rd row to the 1st row         

Add -7/2 times the 2nd row to the 1st row

Then the RREF of B is

1	0	0	2
0	1	0	-7
0	0	1	1

Then the 3^rd column of A^-1 is (2,-7,1)^T.