Question 2. An \( n \times n \) matrix \( A=\left(a_{i j}\right) \) is called lower triangular if for all \( i \) and \( j \) between 1 and \( n \), if \( i<j \) then \( a_{i j}=0 \). For example, \[ \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 2 & 4 & 0 & 0 \\ -3 & 0 & 0 & 0 \\ 0 & 5 & 7 & -8 \end{array}\right) \] is lower triangular. Similarly, an \( n \times n \) matrix \( A=\left(a_{i j}\right) \) is called upper triangular if for all \( i \) and \( j \) between 1 and \( n \), if \( i>j \) then \( a_{i j}=0 \). (a) Prove that the product of two \( n \times n \) lower triangular matrices is lower triangular.
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By definition, for \( A \), if \( i < j \), then \( a_{ij} = 0 \). Similarly, for \( B \), if \( i < j \), then \( b_{ij} = 0 \). 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!
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