# HW12MATH110S17 - HW#12 date March 2 2017 MATH 110 Linear...

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HW #12; date: March 2, 2017MATH 110 Linear Algebrawith Professor Stankova4.2Determinants of Ordern1. (a) False. See Exercise 4.2.1(a).(b) True. It follows from Theorem 4.4.(c) True. It follows from Corollary to Theorem 4.4.(d) True. It follows from Theorem 4.5.(e) False. We should have det(B) =kdet(A).(f) False. We should have det(B) = det(A).(g) False. The identity 2×2 matrix has rank 2 with nonzero determinant.(h) True.LetAbe an upper triangularn×nmatrix.The statementholds forn= 1. Let us use an induction onn. Assumen >1. Considerthe (n-1)×(n-1) matrixBobtained by removing the first row and thefirst column ofA, and leta11, . . . , anndenote the diagonal entries ofA.ThenBis an upper triangular matrix, so by inductiondetB=a22· · ·ann.By the cofactor expansion formula, we havedetA=a11detB=a11· · ·ann,which completes the induction process.3. Add-57of the third row to the second row in the matrix in the lefthand side, and multiply12,13, and17to the first, second and third rowrespectively. Then we obtain the matrix in the right hand side. Thuskshould be 2×3×7 = 42.7. The determinant is-(-1)1230-(-3)0123=-12.10. The determinant is-(-1)2 +i0-11-i+ 3i001-i-2ii2 +i0-1= 4 + 2i.11. The determinant is-(-1)2130-22-101+0131-22301-20231023-111