Find the Inverse of the Resulting Matrix

Problem

[[1,7,1],[2,9,0],[1,8,1]]*[[4.5,0.5,−4.5],[−1,0,1],[3.5,−0.5,−2.5]]

Solution

1. Multiply the two matrices to find the resulting matrix M

M=[[(1)*(4.5)+(7)*(−1)+(1)*(3.5),(1)*(0.5)+(7)*(0)+(1)*(−0.5),(1)*(−4.5)+(7)*(1)+(1)*(−2.5)],[(2)*(4.5)+(9)*(−1)+(0)*(3.5),(2)*(0.5)+(9)*(0)+(0)*(−0.5),(2)*(−4.5)+(9)*(1)+(0)*(−2.5)],[(1)*(4.5)+(8)*(−1)+(1)*(3.5),(1)*(0.5)+(8)*(0)+(1)*(−0.5),(1)*(−4.5)+(8)*(1)+(1)*(−2.5)]]

2. Simplify the arithmetic for each entry of the product.

M=[[4.5−7+3.5,0.5+0−0.5,−4.5+7−2.5],[9−9+0,1+0+0,−9+9+0],[4.5−8+3.5,0.5+0−0.5,−4.5+8−2.5]]

3. Evaluate the sums to obtain the identity matrix.

M=[[1,0,0],[0,1,0],[0,0,1]]

4. Identify the inverse of the identity matrix. Since the product of the two given matrices is the identity matrix I the resulting matrix is I The inverse of the identity matrix is itself.

I(−1)=I

Final Answer

([[1,7,1],[2,9,0],[1,8,1]]*[[4.5,0.5,−4.5],[−1,0,1],[3.5,−0.5,−2.5]])(−1)=[[1,0,0],[0,1,0],[0,0,1]]

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