Prove or disprove the natural conjecture that if a matrix has the property 0 #|arrl >1a1 3=1 3#2 (u>?>I) then the Gaussian elimination without pivoting will preserve this property.
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Matrix A is called strictly column diagonally dominant if |aii| > ∑ |aji|, for j ≠ i. Show that: A is nonsingular. (Hint: Use Gershgorin's theorem.) Show that GEPP does not actually permute any rows, i.e. it is identical to Gaussian elimination without pivoting. (Hint: Show that after one step of Gaussian elimination, the trailing (n-1) x (n-1) submatrix, [Schur complement of a11 in A as per an earlier question], is still diagonally dominant.)
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