Question

Given `A=((-1, 2), (3, 4))` and `B=((-4, 3), (5, -2))`, how do you find `B^-1`?

Answer 1

`B^(-1)=((-2/23,3/23),(5/23,4/23))`

Explanation:

If `B^(-1)` is the Inverse of `B`
then
`color(white)("XXX")BxxB^(-1)=I` where `I` is the identity matrix

Suppose`B^(-1) = ((a,b),(c,d))`

Then
`color(white)("XXX")BxxB^(-1)`
`color(white)("XXX")=((-4,3),(5,2))xx((a,b),(c,d))=((1,0),(0,1))`

`rArr`
`color(white)("XXX")-4a+3c=1color(white)("XXX")`[1]
`color(white)("XXX")-4b+3d=0color(white)("XXX")`[2]
`color(white)("XXXX")5a+2c=0color(white)("XXX")`[3]
`color(white)("XXXX")5b+2d=1color(white)("XXX")`[4]

[1]`xx5 -`[3]`xx4` gives
`color(white)("XXX")23c=5 rarr c=5/23`
and substituting this back into [1] gives
`color(white)("XXX")a=-2/23`

Similarly we can use [2] and [4] to get
`color(white)("XXX")d=4/23`
`color(white)("XXX")b=2/23`

Therefore
`color(white)("XXX")B^(-1)=((-2/23,3/23),(5/23,4/23))`