Question

I need help with inverse matrix, plz
let matrix A= -2 -7 -9
2 5 6
1 3 4
find the second column of inverse matrix A^-1 without computing the other columns.

Answer 1

cofactor, and divide by determinant

Answer 2

and a transpose ... so its really the setup of row 2 cofactors, divided by |A|, transposed

Answer 3

-2 -7 -9
---------
a b c
---------
1 3 4

a = - |-7 -9|
| 3 4|
--------
|A|

b = |-2 -9|
| 1 4|
--------
|A|

c = - |-2 -7|
| 1 3|
--------
|A|


[a b c] is the column vector of A^-1

Answer 4

- (-45-36-56)
-2 -7 -9 -2 -7
2 5 6 2 5
1 3 4 1 3
-40-42-54

which of course is eqaul to 1 for the determinant ...

Answer 5

so the 2nd column of the inverse matrix is related to 2nd row of the A matrix?

Answer 6

yes

Answer 7

in inverse can be formulated from taking the cofactors of the given matrix, dividing by the determinant, and transposing

Answer 8

|a b|
|c d|

a to |d| = d
b to -|c| = -c
c to -|b| = -b
d to |a| = a

to give us a cofactor matrix of:

| d -c|
|-b a|

divide the elements by the determinant; lets assume 1 for simplicity here; and transpose

| d -b|
|-c a|

this would be the inverse matrix of a 2x2

Answer 9

since the determinant of yours is graciously 1 ... this simplifies to

a = - |-7 -9| = 1
| 3 4|


b = |-2 -9| = 1
| 1 4|


c = - |-2 -7| = -1
| 1 3|

Answer 10

thank you so much

Answer 11

does it make sense?

Answer 12

sure