Simplify the Matrix

Problem

[[1,−1,0],[0,1,0],[0,0,1]]*[[0,1,0],[1,0,0],[0,0,1]]*[[1,0,0],[0,1,0],[0,0,−2]]

Solution

1. Multiply the first two matrices together using the standard row-by-column rule.

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

2. Simplify the resulting intermediate matrix.

[[−1,1,0],[1,0,0],[0,0,1]]

3. Multiply this intermediate matrix by the third matrix.

[[−1,1,0],[1,0,0],[0,0,1]]*[[1,0,0],[0,1,0],[0,0,−2]]=[[(−1)*(1)+(1)*(0)+(0)*(0),(−1)*(0)+(1)*(1)+(0)*(0),(−1)*(0)+(1)*(0)+(0)*(−2)],[(1)*(1)+(0)*(0)+(0)*(0),(1)*(0)+(0)*(1)+(0)*(0),(1)*(0)+(0)*(0)+(0)*(−2)],[(0)*(1)+(0)*(0)+(1)*(0),(0)*(0)+(0)*(1)+(1)*(0),(0)*(0)+(0)*(0)+(1)*(−2)]]

4. Simplify the final entries to obtain the resulting matrix.

[[−1,1,0],[1,0,0],[0,0,−2]]

Final Answer

[[1,−1,0],[0,1,0],[0,0,1]]*[[0,1,0],[1,0,0],[0,0,1]]*[[1,0,0],[0,1,0],[0,0,−2]]=[[−1,1,0],[1,0,0],[0,0,−2]]

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