Find the Inverse of the Resulting Matrix

Problem

[[−0.8,0.1,0.1],[0.6,0.5,0.2],[0.2,0.4,−0.2]]*[[0],[0],[0]]

Solution

1. Perform matrix multiplication by multiplying the 3×3 matrix by the 3×1 zero column vector.

2. Calculate the product where each entry is the dot product of a row from the first matrix and the zero vector.

[[−0.8*(0)+0.1*(0)+0.1*(0)],[0.6*(0)+0.5*(0)+0.2*(0)],[0.2*(0)+0.4*(0)+−0.2*(0)]]=[[0],[0],[0]]

3. Determine invertibility of the resulting 3×1 matrix. A matrix must be square (having the same number of rows and columns) to have an inverse.

4. Check the determinant condition. Even if we consider the resulting vector as a component of a system, a zero matrix or a non-square matrix does not possess an inverse. Specifically, the zero vector is singular and cannot be inverted.

Final Answer

The inverse does not exist because the resulting matrix is a zero vector.

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