Question

An $n imes n$ matrix A is called upper triangular if $a_{ij} = 0$ whenever $i > j$, and is called lower triangular if $a_{ij} = 0$ whenever $i < j$. (a) Suppose that A, B $in M_n(mathbb{F})$ are both upper triangular. Prove that AB is also upper triangular. What are the diagonal entries of AB? (b) Do the same for lower triangular matrices. Warning: Don't be tricked by this into thinking that AB = BA for triangular matrices!

          An $n 	imes n$ matrix A is called upper triangular if $a_{ij} = 0$ whenever $i > j$, and is called lower triangular if $a_{ij} = 0$ whenever $i < j$. (a) Suppose that A, B $in M_n(mathbb{F})$ are both upper triangular. Prove that AB is also upper triangular. What are the diagonal entries of AB? (b) Do the same for lower triangular matrices. Warning: Don't be tricked by this into thinking that AB = BA for triangular matrices!

An n imes n matrix A is called upper triangular if aij = 0 whenever i > j, and is called lower triangular if aij = 0 whenever i < j. (a) Suppose that A, B in Mn(mathbbF) are both upper triangular. Prove that AB is also upper triangular. What are the diagonal entries of AB? (b) Do the same for lower triangular matrices. Warning: Don't be tricked by this into thinking that AB = BA for triangular matrices!

Added by Christopher A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

04/26/2023

Step 1

To do this, we need to show that the (i, j) entry of AB is 0 whenever i > j. Recall that the (i, j) entry of AB is given by the sum of the products of the corresponding entries in the i-th row of A and the j-th column of B, i.e., (AB)_{ij} = \sum_{k=1}^n Show more…

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An n x n matrix A is called upper triangular if aij = 0 whenever i > j, and is called lower triangular if aij = 0 whenever i < j: (a) Suppose that A,B 6 Mn(F) are both upper triangular Prove that AB is also upper triangular: What are the diagonal entries of AB? (b) Do the same for lower triangular matrices Warning: Dont be tricked by this into thinking that AB = BA for triangular matrices!