Find the Inverse

Problem

[[1,1,1,1,1,1],[0,0.3,0.53,0.76,0.99,1.27],[−0.3,0,0.23,0.46,0.69,0.97],[−0.53,−0.23,0,0.23,0.46,0.74],[−0.76,−0.46,−0.23,0,0.23,0.51],[−0.99,−0.69,−0.46,−0.23,0,0.28]]^(−1)

Solution

1. Identify the structure of the matrix. Let the given matrix be A We observe the relationships between the rows.

2. Calculate the difference between consecutive rows. Notice that for the first five rows:

R*o*(w_2)−R*o*(w_3)=[[0.3,0.3,0.3,0.3,0.3,0.3]]

R*o*(w_3)−R*o*(w_4)=[[0.23,0.23,0.23,0.23,0.23,0.23]]

R*o*(w_4)−R*o*(w_5)=[[0.23,0.23,0.23,0.23,0.23,0.23]]

R*o*(w_5)−R*o*(w_6)=[[0.23,0.23,0.23,0.23,0.23,0.23]]

3. Determine linear dependence. Since R*o*(w_2)−R*o*(w_3)=0.3⋅R*o*(w_1) we can express one row as a linear combination of others:

R*o*(w_3)=R*o*(w_2)−0.3⋅R*o*(w_1)

4. Conclude the invertibility. Because at least one row is a linear combination of other rows, the rows are linearly dependent.

5. Evaluate the determinant. A matrix with linearly dependent rows has a determinant of zero.

det(A)=0

6. State the result. A matrix with a determinant of zero is singular and does not have an inverse.

Final Answer

[[1,1,1,1,1,1],[0,0.3,0.53,0.76,0.99,1.27],[−0.3,0,0.23,0.46,0.69,0.97],[−0.53,−0.23,0,0.23,0.46,0.74],[−0.76,−0.46,−0.23,0,0.23,0.51],[−0.99,−0.69,−0.46,−0.23,0,0.28]]^(−1)=Undefined (Singular Matrix)

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