1. Show that if A is an upper triangular matrix and invertible then A^-1 is also upper triangular?

2. Show that if A is an lower triangular matrix and invertible then A^-1 is also lower triangular

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  • 6 years ago
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    Picture the product of A with its inverse. A is a upper triangulaer matrix (of size n) and B (the inverse) is a as yet matrix with elements B(i,j). Their product is the identity matrix.

    Focus on the last minus -1 element of the last row of the identity matrix: its value 0. It is created by the sum of A(n,j)*B(j,n-1) = 0

    As all elements of the last row of A are 0 (except for the last), this sum = A(n,n).B(n,n-1) = 0 As A(n,n_ not equal to zero, so B(n-1,n) must be. So you can work out from B(n-1,n) until B(2,1), that all the elements in the lower part must be zero.

    The same line of reasoning applies for the lower triangular matrix.

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