Matrix Determinant Calculation

The matrix is given by \(M_k = \begin{bmatrix} k & k-1 \\ k-1 & k \end{bmatrix}\). The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is \(ad - bc\). Therefore, the determinant of \(M_k\), denoted as \(|M_k|\), is calculated as: \( |M_k| = (k \times k) - ((k-1) \times (k-1)) \) \( |M_k| = k^2 - (k^2 - 2k + 1) \) \( |M_k| = k^2 - k^2 + 2k - 1 \) \( |M_k| = 2k - 1 \)

Summation of Determinants

The question asks for the sum \(S = |M_1| + |M_2| + |M_3| + \dots + |M_{50}|\). Substituting the expression for \(|M_k|\), we get: \( S = \sum_{k=1}^{50} |M_k| = \sum_{k=1}^{50} (2k - 1) \)

Applying Summation Formula

The expression \(\sum_{k=1}^{n} (2k - 1)\) represents the sum of the first \(n\) odd natural numbers, which is known to be equal to \(n^2\). In this case, \(n = 50\).

Final Calculation

Using the formula, the sum \(S\) is: \( S = 50^2 \) \( S = 2500 \)