Find the Inverse

Problem

[[0.5,0,0,0,0,0,1,0,0,0],[0.866,0,0,0,0,0,0,−1,0,0],[−0.5,0.101,0,0,−0.866,0,0,0,0,0],[−0.866,0.101,0,0,0.5,1,0,0,0,0],[0,−0.101,0,0,0,0,0,0,0,800],[0,−0.101,−1,0,0,0,0,0,0,0],[0,0,0,0,0,−1,0,0,0,250],[0,0,1,−1,0,0,0,0,0,0],[0,0,0,0,0.5,0,0,0,0,0],[0,0,0,1,0.866,0,0,1,0,0]](−1)

Solution

1. Identify the structure of the 10×10 matrix A Note that many columns and rows contain only one or two non-zero entries, suggesting we can solve the system A*X=I or use row reduction.

2. Solve for the variables by setting up the equation A*x=(e_j) for each basis vector. For example, from row 9, 0.5*(x_5)=(y_9)⇒(x_5)=2*(y_9)

3. Substitute known values into other equations. From row 5, −0.101*(x_2)+800*(x_10)=(y_5) From row 7, −(x_6)+250*(x_10)=(y_7) From row 10, (x_4)+0.866*(x_5)+(x_8)=(y_10)

4. Isolate each variable (x_1) through (x_10) in terms of the identity components (y_1) through (y_10) to determine the entries of the inverse matrix A(−1)

5. Calculate the numerical values. For instance, (x_10) is found to be 0 in most columns except where (y_5) or (y_6) are involved, but since the last column of the original matrix is mostly zeros except for rows 5 and 7, we find the matrix is singular (determinant is zero) because the 9th column of the original matrix consists entirely of zeros.

6. Verify the column of zeros. Observing the 9th column of the input matrix: all entries are 0

7. Conclude that a matrix with a column of all zeros has a determinant of 0 and is therefore not invertible.

Final Answer

[[0.5,0,0,0,0,0,1,0,0,0],[0.866,0,0,0,0,0,0,−1,0,0],[−0.5,0.101,0,0,−0.866,0,0,0,0,0],[−0.866,0.101,0,0,0.5,1,0,0,0,0],[0,−0.101,0,0,0,0,0,0,0,800],[0,−0.101,−1,0,0,0,0,0,0,0],[0,0,0,0,0,−1,0,0,0,250],[0,0,1,−1,0,0,0,0,0,0],[0,0,0,0,0.5,0,0,0,0,0],[0,0,0,1,0.866,0,0,1,0,0]](−1)=Undefined (Singular Matrix)

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