More- over, if the partition is in fact an all-square partition and A, B, and D are all invertible, then (3.2) We can assume that the matrix A is upper triangular and invertible, since \(\displaystyle A^{-1}=\frac{1}{det(A)}\cdot adj(A)\) We can prove that \(\displaystyle A^{-1}\) is upper triangular by showing that the adjoint is upper triangular or that the matrix of cofactors is lower triangular. Suppose the n × n matrix R is upper triangular and invertible, i.e., its diagonal entries are all nonzero. Clearly, the inverse of a block upper triangular matrix is block upper triangular only in the square diagonal partition. (a) if U is upper triangular and invertible then U^-1 is upper triangular. It fails the test in Note 5, because ad bc equals 2 2 D 0. An LU factorization refers to the factorization of A, with proper row and/or column orderings or permutations, into two factors – a lower triangular matrix L and an upper triangular matrix U: =. Taking transposes leads immediately to: Corollary If the inverse L 1 of an lower triangular matrix L exists, Show that R1 is also upper triangular. It's obvious that upper triangular matrix is also a row echelon matrix. Proposition If a lower (upper) triangular matrix is invertible, then its inverse is lower (upper) triangular. Hint. Let A and B be upper triangular matrices of size nxn. (b) The inverse of a unit lower triangular matrix is unit lower triangular (c) The product of two upper or (two lower triangular) matrices is upper or (lower) triangular It goes like this: the triangular matrix is a square matrix where all elements below the main diagonal are zero. An example is the 4 4 matrix 4 5 10 1 0 7 1 1 0 0 2 0 0 0 0 9 . The inverse element of the matrix [begin{bmatrix} 1 & x & y \ 0 &1 &z \ 0 & 0 & 1 end{bmatrix}] is given by [begin{bmatrix} 1 & -x & xz-y \ 0 & 1 & -z \ 0 & 0 & 1 end{bmatrix}.] Inverse of matrix : A square matrix of order {eq}n \times n{/eq} is known as an upper triangular matrix if all the elements below principle diagonal elements are zero. Let A be a square matrix. 11.7 Inverse of an upper triangular matri. Let [math]b_{ij}[/math] be the element in row i, column j of B. Let A be a n n upper triangular matrix with nonzero diagonal entries. For a proof, see the post The inverse matrix of an upper triangular matrix with variables. The proof for upper triangular matrices is similar (replace columns with rows). Use back substitution to solve Rsk-en for k 1, , n, and argue that (sk)i -0 for i > k. The inverse of a triangular matrix is triangular. It's actually called upper triangular matrix, but we will use it. In the next slide, we shall prove: Theorem If the inverse U 1 of an upper triangular matrix U exists, then it is upper triangular. In this problem, you will Example of upper triangular matrix: 1 0 2 5 0 3 1 3 0 0 4 2 0 0 0 3 A triangular matrix (upper or lower) is invertible if and only if no element on its principal diagonal is 0. Solving Linear Equations Note 6 A diagonal matrix has an inverse provided no diagonal entries are zero: If A D 2 6 4 d1 dn 3 7 5 then A 1 D 2 6 4 1=d1 1=dn 3 7 5: Example 1 The 2 by 2 matrix A D 12 12 is not invertible. In the lower triangular matrix all elements above the diagonal are zero, in the upper triangular matrix, all the elements below the diagonal are zero. In general this is not true for the square off-diagonal partition. Let [math]a_{ij}[/math] be the element in row i, column j of A. An upper triangular matrix is a square matrix in which the entries below the diagonal are all zero, that is, a ij = 0 whenever i > j. 82 Chapter 2.

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