What is the sum of the entries in the second column of the inverse of the following matrix? HINT: You don't need to find the entire inverse to answer this question.
\[
\left[\begin{array}{cccc}
1 & -1 & -5 & -1 \\
1 & 1 & 1 & 1 \\
-4 & -1 & 1 & 3 \\
2 & -3 & -1 & -1
\end{array}\right]
\]
Answer
\(\boxed{1}\)
Step 1 ：Set up a system of linear equations using the property that the product of a matrix and its inverse is the identity matrix: \(\begin{bmatrix} -1b_{11} + 1b_{21} - 5b_{31} - 1b_{41} = 0 \\ 1b_{11} + 1b_{21} + 1b_{31} + 1b_{41} = 1 \\ -4b_{11} - 1b_{21} + 1b_{31} + 3b_{41} = 0 \\ 2b_{11} - 3b_{21} - 1b_{31} - 1b_{41} = 0 \end{bmatrix}\)
Step 2 ：Solve the system of linear equations to find the sum \(b_{12} + b_{22} + b_{32} + b_{42} = 1\)
Step 3 ：\(\boxed{1}\)
