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Unformatted text preview: Theorem 1.7.1: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular. (c) A triangular matrix is invertible if and only if its diagonal entries are all non zero. (d) The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular. Symmetric Matrices: A square matrix A is symmetric if A = A T . Example: 1. Zero matrix and identity matrix are symmetric (any diagonal matrix is sym- metric) 2. 6 8 5 8 2- 1 5- 1 is a symmetric matrix Theorem 1.7.2, 1.7.3, 1.7.4: If A and B are symmetric matrices with the same size, and if k is any scalar, then: (a) A T is symmetric (b) A + B and A- B are symmetric (c) kA is symmetric (d) The product of two symmetric matrices is symmetric if and only if the matrices commute. (e) If A is an invertible symmetric matrix, then A- 1 is symmetric. Chapter 2, Determinants Section 2.1, Determinants by Cofactor Expansion Importance: We will use determinants for deciding whether a matrix is invert- ible or not. Also they can be used to write a formula to find inverse of a matrix. Another usage is finding solutions of a linear system of equations. So far we were using a method (forming augmented matrix and row reduction) where we found solu-...
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- Spring '11