Question

How do you find the inverse of `A=``((-3, -3, -4), (0, 1, 1), (4, 3, 4))`?

Answer 1

The answer is `=((1,0,1),(4,4,3),(-4,-3,-3))`

Explanation:

We start by calculating the determinant of matrix `A` to see if it's invertible

`det A=|(-3,-3,-4),(0,1,1),(4,3,4)|`

`=-3*|(1,1),(3,4)|+3*|(0,1),(4,4)|-4*|(0,1),(4,3)|`

`=(-3*1)+(3*-4)-4(-4)`

`=-3-12+16=1`

Therefore,

`detA!=0`, the matrix is invertible

We start by calculating the matrix of cofactors

`C=((|(1,1),(3,4)|,-|(0,1),(4,4)|,|(0,1),(4,4)|),(-|(-3,-4),(3,4)|,|(-3,-4),(4,4)|, -|(-3,-3),(4,3)|),(|(-3,-4),(1,1)|,-|(-3,-4),(0,1)|,|(-3,-3),(0,1)|))`

`C=((1,4,-4),(0,4,-3),(1,3,-3))`

Now, we determine the transpose of `C`

`C^T=((1,0,1),(4,4,3),(-4,-3,-3))`

So,

`A^-1=C^T/(detA)`

`A^-1=C=((1,0,1),(4,4,3),(-4,-3,-3))`

Verification

`A*A^-1=((-3,-3,-4),(0,1,1),(4,3,4))*((1,0,1),(4,4,3),(-4,-3,-3))`

`=((1,0,0),(0,1,0),(0,0,1))=I`