Find the Inverse of the Resulting Matrix

Problem

([[3,1/2,5],[1,−1,3]]+[[2,−5/2,0],[0,2,−3]])*[[1,0,−1],[1,2,0],[0,3,0]]

Solution

1. Add the two matrices inside the parentheses by summing their corresponding entries.

[[3,1/2,5],[1,−1,3]]+[[2,−5/2,0],[0,2,−3]]=[[5,−2,5],[1,1,0]]

2. Multiply the resulting 2×3 matrix by the 3×3 matrix.

[[5,−2,5],[1,1,0]]*[[1,0,−1],[1,2,0],[0,3,0]]=[[(5)*(1)+(−2)*(1)+(5)*(0),(5)*(0)+(−2)*(2)+(5)*(3),(5)*(−1)+(−2)*(0)+(5)*(0)],[(1)*(1)+(1)*(1)+(0)*(0),(1)*(0)+(1)*(2)+(0)*(3),(1)*(−1)+(1)*(0)+(0)*(0)]]

3. Simplify the arithmetic to find the product matrix A

A=[[3,11,−5],[2,2,−1]]

4. Identify the dimensions of the resulting matrix. The matrix A is a 2×3 matrix.

5. Determine if an inverse exists. A matrix must be square (having the same number of rows and columns) to have an inverse. Since this matrix is 2×3 it is not square.

Final Answer

The resulting matrix *[[3,11,−5],[2,2,−1]]* has no inverse because it is not square.

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