Find the Inverse of the Resulting Matrix

Problem

[[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]*[[81],[8],[9]]

Solution

1. Multiply the matrix by the vector to find the resulting matrix B

B=[[(0.30⋅81)+(0.20⋅8)+(0.30⋅9)],[(0.40⋅81)+(0.30⋅8)+(0.30⋅9)],[(0.30⋅81)+(0.30⋅8)+(0.20⋅9)]]

2. Calculate the individual sums for each row.

B=[[24.3+1.6+2.7],[32.4+2.4+2.7],[24.3+2.4+1.8]]

3. Simplify the vector to obtain the final resulting matrix.

B=[[28.6],[37.5],[28.5]]

4. Identify the inverse operation. Since the result is a 3×1 matrix (a column vector), it is not a square matrix.

5. Conclude that a non-square matrix does not have a standard multiplicative inverse. In linear algebra, the inverse A(−1) is only defined for square matrices where the determinant is non-zero.

Final Answer

The resulting matrix *[[28.6],[37.5],[28.5]]* has no inverse because it is not square.

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