8/11/22, 3:41 AM a. Show that the product of two $n \times n$ lower triangula | Quizlet 1 Upgrade Expert solutions Question a. Show that the product of two n × n lower triangular matrices is lower triangular. b. Show that the product of two n × n upper triangular matrices is upper triangular. c. Show that the inverse of a nonsingular n × n lower triangular matrix is lower triangular. Solution Verified Step 1 1 of 3 = [lij ] is lower -triangular matrix if lij = 0 for each i = 1, 2, … , j − 1. Let A and B be two n × n lower triangular (a) A square matrix L matrices. So aij = bij = 0 Now AB for each i = 1, 2, … , j − 1 n = [cij ] is given by cij = ∑k=1 aik bkj so if i ≤ j − 1 then j−1 n n ∑ aik bkj = ∑ aik bkj + ∑ aik bkj k=1 k=1 k=j = 0 since k ≤ j − 1 and on second part of above sum aik = 0 since i ≤ j − 1 < k . Thus, we have On first part, bkj j−1 n n ∑ aik bkj = ∑ aik bkj + ∑ aik bkj = 0 k=1 i.e. cij k=1 k=j = 0 if i = 1, 2, … , j − 1. Thus AB be lower triangular as well. https://quizlet.com/explanations/questions/a-show-that-the-product-of-two-n-times-n-lower-triangular-matrices-is-lower-triangular-b-show-that-the-… 1/5 8/11/22, 3:41 AM a. Show that the product of two $n \times n$ lower triangula | Quizlet Step 2 2 of 3 (b) An upper-triangular n × n matrix U = [uij ] has, for each j = 1, 2, … , n, the entries uij = 0, for each i = j + 1, j + 2, … , n Suppose A and B be two n × n upper triangular matrices. If A = [aij ] and B = [bij ] then ⎡ ⎤ AB = (∑ aik bkj ) ⎣ k=1 ⎦ ij Now suppose that i k ≥ j + 1 then bkj n ≥ j + 1 then if k ≤ j + 1 then aik = 0 and if = 0 so we get j n n ∑ aik bkj = ∑ aik bkj + ∑ aik bkj = 0 + 0 k=1 k=1 k=j+1 Thus the matrix AB is also upper triangular. https://quizlet.com/explanations/questions/a-show-that-the-product-of-two-n-times-n-lower-triangular-matrices-is-lower-triangular-b-show-that-the-… 2/5 8/11/22, 3:41 AM a. Show that the product of two $n \times n$ lower triangula | Quizlet Step 3 3 of 3 (c) Suppose that A is non-singular lower triangular matrix and if A−1 is not lower triangular then there exists element a−1 lm , where 1 ≤ l, m ≤ n and l ≤ m. Now, take element lk -th element of A−1 A which is n k −1 ∑ a−1 lj ajk = ∑ alj ajk j=1 j=1 = 1, 2, … , n then we have If we take k for each k k ∑ a−1 lj ajk j=1 If we exclude the case that l 1 ={ 0 if l = k otherwise = k then we get get k ∑ a−1 lj ajk = 0, for each k = 1, 2, … , l − 1, l + 1, … , n j=1 Now we know that the columns of A is linearly independent since A is −1 = 0 with m ≥ l, this contradicts that columns of A are linearly independent i.e. A is non-singular. Thus, A−1 must also be lower triangular. non-singular, and we know that alm Recommended textbook solutions https://quizlet.com/explanations/questions/a-show-that-the-product-of-two-n-times-n-lower-triangular-matrices-is-lower-triangular-b-show-that-the-… 3/5 8/11/22, 3:41 AM a. Show that the product of two $n \times n$ lower triangula | Quizlet Numerical Analysis 3rd Edition Timothy Sauer 383 solutions Numerical Analysis 2nd Edition Timothy Sauer 385 solutions Numerical Analysis 9th Edition J. Douglas Faires, Richard L. Burden 873 solutions Numerical Analysis 9th Edition J. Douglas Faires, Richard L. Burden 873 solutions Related questions COMPUTER SCIENCE Prove that the product of two lower-triangular matrices is lower-triangular. LINEAR ALGEBRA Is the product of two lower triangular ma triangular matrix as well? Explain vour a LINEAR ALGEBRA LINEAR ALGEBRA Prove that a product of unit lower triangular matrices is unit lower triangular. Prove that the product of two upper triangular n×n https://quizlet.com/explanations/questions/a-show-that-the-product-of-two-n-times-n-lower-triangular-matrices-is-lower-triangular-b-show-that-the-… 4/5 8/11/22, 3:41 AM a. Show that the product of two $n \times n$ lower triangula | Quizlet matrices is upper triangular. Privacy Terms https://quizlet.com/explanations/questions/a-show-that-the-product-of-two-n-times-n-lower-triangular-matrices-is-lower-triangular-b-show-that-the-… 5/5