Find the Inverse [[0,0,1],[0,1,0],[-1,0,0]]

Problem

[[0,0,1],[0,1,0],[−1,0,0]]

Solution

1. Set up the augmented matrix by placing the identity matrix I to the right of the given matrix A

[[0,0,1,|,1,0,0],[0,1,0,|,0,1,0],[−1,0,0,|,0,0,1]]

2. Swap the first and third rows ((R_1)↔(R_3) to move a non-zero element to the top-left pivot position.

[[−1,0,0,|,0,0,1],[0,1,0,|,0,1,0],[0,0,1,|,1,0,0]]

3. Multiply the first row by −1 (−1*(R_1)→(R_1) to make the leading coefficient of the first row equal to 1

[[1,0,0,|,0,0,−1],[0,1,0,|,0,1,0],[0,0,1,|,1,0,0]]

4. Identify the resulting matrix on the right side as the inverse, since the left side is now the identity matrix.

A(−1)=[[0,0,−1],[0,1,0],[1,0,0]]

Final Answer

[[0,0,1],[0,1,0],[−1,0,0]](−1)=[[0,0,−1],[0,1,0],[1,0,0]]

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