Find the Inverse of the Resulting Matrix

Problem

([[3,2],[−1,0],[0,−1]]*[[−1,0,−1],[2,1,0]]*[[1,1,0],[2,1,0],[3,0,1]])(−1)

Solution

1. Multiply the first two matrices to find the product A⋅B

[[3,2],[−1,0],[0,−1]]*[[−1,0,−1],[2,1,0]]=[[(3)*(−1)+(2)*(2),(3)*(0)+(2)*(1),(3)*(−1)+(2)*(0)],[(−1)*(−1)+(0)*(2),(−1)*(0)+(0)*(1),(−1)*(−1)+(0)*(0)],[(0)*(−1)+(−1)*(2),(0)*(0)+(−1)*(1),(0)*(−1)+(−1)*(0)]]

[[3,2],[−1,0],[0,−1]]*[[−1,0,−1],[2,1,0]]=[[1,2,−3],[1,0,1],[−2,−1,0]]

2. Multiply the resulting matrix by the third matrix to find the final product M

M=[[1,2,−3],[1,0,1],[−2,−1,0]]*[[1,1,0],[2,1,0],[3,0,1]]

M=[[(1)*(1)+(2)*(2)+(−3)*(3),(1)*(1)+(2)*(1)+(−3)*(0),(1)*(0)+(2)*(0)+(−3)*(1)],[(1)*(1)+(0)*(2)+(1)*(3),(1)*(1)+(0)*(1)+(1)*(0),(1)*(0)+(0)*(0)+(1)*(1)],[(−2)*(1)+(−1)*(2)+(0)*(3),(−2)*(1)+(−1)*(1)+(0)*(0),(−2)*(0)+(−1)*(0)+(0)*(1)]]

M=[[−4,3,−3],[4,1,1],[−4,−3,0]]

3. Calculate the determinant of matrix M using expansion along the third row.

det(M)=−4*|[3,−3],[1,1]|−(−3)*|[−4,−3],[4,1]|+0*|[−4,3],[4,1]|

det(M)=−4*(3−(−3))+3*(−4−(−12))

det(M)=−4*(6)+3*(8)=−24+24=0

4. Conclude based on the determinant. Since the determinant of the resulting matrix is 0 the matrix is singular.

Final Answer

([[3,2],[−1,0],[0,−1]]*[[−1,0,−1],[2,1,0]]*[[1,1,0],[2,1,0],[3,0,1]])(−1)=Undefined (Matrix is singular)

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