If M_k= bmatrix k & k-1 k-1 & k bmatrix where k is a natural number, then what is |M₁|+|M₂|+|M₃|+ +|M₅₀| equal to ?

Step-by-step solution

Given M_k = bmatrix k & k-1 k-1 & k bmatrix The determinant of M_k is given by: |M_k| = k^2 - (k-1)^2 |M_k| = k^2 - (k^2 - 2k + 1) = 2k - 1 The required sum is: |M₁| + |M₂| + |M₃| + + |M₅₀| = _ k=1 ⁵⁰ (2k - 1) This represents the sum of the first 50 odd natural numbers, which forms an arithmetic progression: 1 + 3 + 5 + + 99 The sum of an arithmetic progression is given by n 2 (a + l) , where n = 50 , a = 1 , and l = 99 : Sum = 50 2 (1 + 99) = 25 100 = 2500 Answer: 2500