Question

Consider the 4 × 4 matrix \(M=\begin{pmatrix} 11 & 10 & 10 & 10 \\ 10 & 11 & 10 & 10 \\ 10 & 10 & 11 & 10 \\ 10 & 10 & 10 & 11 \end{pmatrix}\). Then the value of the determinant of M is equal to _________.

Answer 1

Correct Answer: 41

The matrix \( M \) is a 4x4 matrix given by: 

\(M=\begin{pmatrix} 11 & 10 & 10 & 10 \\ 10 & 11 & 10 & 10 \\ 10 & 10 & 11 & 10 \\ 10 & 10 & 10 & 11 \end{pmatrix}\)

To find the determinant of \( M \), we observe that \( M \) can be expressed as \( M = A + 10B \), where \( A \) is the identity matrix and \( B \) is a matrix with all elements 1. Specifically, \( A \) and \( B \) are:

\(A=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\)
\(B=\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}\)

Therefore, the determinant of \( M \) is \( \det(M) = \det(A + 10B) \). Using the Sherman-Morrison formula for the determinant of a matrix in the form of \( A + uv^T \), we can compute the determinant as follows:

1. Compute \( \det(A) = 1 \) since \( A \) is the identity matrix.
2. Recognize that the matrix \( B \) contributes to making \( A + 10B \) a rank-1 adjustment of \( A \).
3. Use the Sherman-Morrison formula: \(\det(A + uv^T) = (1 + v^T A^{-1} u) \det(A)\), where \( u \) and \( v \) are column vectors comprised of ones, multiplied by 10 in this case. 
4. Since all elements of \( B \) are identical, each column sum of \( B \) is 4, implying \( v^T A^{-1} u = 4\).

The determinant is then calculated as:
\(\det(M) = 1 \times (1 + 10 \times 4) = 1 + 40 = 41\).