Find the Inverse of the Resulting Matrix

Problem

([[1,0,0],[0,1,0],[0,0,1]]−[[0.30,0.20,0.30],[0.30,0.20,0.40],[0.20,0.30,0.30]])*[[52],[74],[29]]

Solution

1. Perform matrix subtraction to find the matrix inside the parentheses.

[[1,0,0],[0,1,0],[0,0,1]]−[[0.30,0.20,0.30],[0.30,0.20,0.40],[0.20,0.30,0.30]]=[[0.7,−0.2,−0.3],[−0.3,0.8,−0.4],[−0.2,−0.3,0.7]]

2. Multiply the resulting matrix by the column vector.

[[0.7,−0.2,−0.3],[−0.3,0.8,−0.4],[−0.2,−0.3,0.7]]*[[52],[74],[29]]=[[(0.7)*(52)+(−0.2)*(74)+(−0.3)*(29)],[(−0.3)*(52)+(0.8)*(74)+(−0.4)*(29)],[(−0.2)*(52)+(−0.3)*(74)+(0.7)*(29)]]

3. Calculate the arithmetic for each row of the vector.

Row 1: *36.4−14.8−8.7=12.9

Row 2: −15.6+59.2−11.6=32

Row 3: −10.4−22.2+20.3=−12.3

4. Identify the resulting matrix as a 3×1 column vector.

[[12.9],[32],[−12.3]]

5. Determine the inverse of the resulting matrix. A non-square matrix (such as a 3×1 vector) does not have a standard multiplicative inverse.

Final Answer

([[1,0,0],[0,1,0],[0,0,1]]−[[0.30,0.20,0.30],[0.30,0.20,0.40],[0.20,0.30,0.30]])*[[52],[74],[29]]=[[12.9],[32],[−12.3]]

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