An engineer is applying Gaussian Elimination to solve a syst

An engineer is applying Gaussian Elimination to solve a system of linear equations with a 401x401 coefficient matrix. At the end of 246 steps of forward elimination, there are at least ____ zeros in the matrix.


0: Forward elimination works its way through the matrix column by column and reduces the lower part of the matrix to zeros. After the first step, the first column will only have one non-zero coefficient left (a₁₁). After the second step, the second column will only have two non-zero coefficients left (a₁₂ and a₂₂\'). Follow this thinking excercise, try to observe patterns and derive a formula for the number of zeros in the lower half of the matrix.

Answer: ...............

Solution

The size of matrix is 401 X 401

Hence after first forward elimination, there will be only one non zero coefficient (a11), hence the minimum number of zeroes in the first column will be

401 - 1 = 400 zeroes

Second column will contain (401 - 2) = 399 zeroes (a12 and a22\')

Third column will contain (401 - 3) = 398 zeroes (a13, a23 and a33)

Hence minimum number of zeroes

=> (401-1) + (401-2) + (401-3) + ... + (401-246)

=> 401 * 246 - (246 * 247)/2

=> 68265