# How do you find the inverse of A=((0, 1, 2), (1, 3, 5), (-2, -3, -5))?

Feb 16, 2016

${A}^{- 1} = \left(\begin{matrix}0 & - 1 & - 1 \\ - 5 & 4 & 2 \\ 3 & - 2 & - 1\end{matrix}\right)$

#### Explanation:

Expand $A$ with an identity matrix appended to its right side:

((0,1,2,"|",1,0,0), (1,3,5,"|",0,1,0), (-2,-3,-5,"|",0,0,1))

Our objective will be to perform standard row operations which will convert this matrix into a matrix with the identity matrix on the left side

Exchanging Row 1 and Row 2
((1,3,5,"|",0,1,0), (0,1,2,"|",1,0,0), (-2,-3,-5,"|",0,0,1))

Row 3 = ($2 \times$ Row 1) plus Row 3
((1,3,5,"|",0,1,0), (0,1,2,"|",1,0,0), (0,3,5,"|",0,2,1))

Row 1 = Row 1 minus ($3 \times$ Row 2)
((1,0,-1,"|",-3,1,0), (0,1,2,"|",1,0,0), (0,3,5,"|",0,2,1))

Row 3 = ($3 \times$ Row 2) minus Row 3
((1,0,-1,"|",-3,1,0), (0,1,2,"|",1,0,0), (0,0,1,"|",3,-2,-1))

Row 1 = Row 1 plus Row 3
((1,0,0,"|",0,-1,-1), (0,1,2,"|",1,0,0), (0,0,1,"|",3,-2,-1))

Row 2 = Row 2 minus ($2 \times$ Row 3)
((1,0,0,"|",color(green)(0),color(green)(-1),color(green)(-1)), (0,1,0,"|",color(green)(-5),color(green)(4),color(green)(2)), (0,0,1,"|",color(green)(3),color(green)(-2),color(green)(-1)))

The half of the matrix on the right side is the inverse of the original matrix.

You should check this result to ensure:
$A \times {A}^{- 1} = I$
that is
$\left(\begin{matrix}0 & 1 & 2 \\ 1 & 3 & 5 \\ - 2 & - 3 & - 5\end{matrix}\right) \times \left(\begin{matrix}0 & - 1 & - 1 \\ - 5 & 4 & 2 \\ 3 & - 2 & - 1\end{matrix}\right) = \left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right)$