Use 10 factorization of A to solve the system Ax b show tha

Use 10 factorization of A to solve the system Ax = b, show that the inverse of an invertible upper triangular matrix an upper triangular matrix Let V be the vector space spanned by the functions, Find the matrix A of the operator relative to the basis Find the eigenvalues of A the matrix A diagonalizable?

Solution

2)

Here we use math induction. It is obvious that the inverse of a 1×1 lower triangular matrix is lower triangular. Now assume that the inverse of n × n lower triangular matrix is lower triangular, we prove it is also true for (n + 1) × (n + 1) matrix B. Notice an (n + 1) × (n + 1) lower triangular matrix B is obtained by adding a possibily non-zero row to the bottom to a n×n lower triangular matrix A (and a zero column to its right). We will prove B¹ can be obtained by adding a possibly non-zero row to the bottom of A¹ and a zero column to the right of A¹ . We use the each row of B, from row 1 to row n, to multiply the last column of B^{1 ,} and we will find that the last column of B1 is zero (excluing the last entry of this column). This prove that B¹ is also lower triangular. Use the last row of B to multiply each column of B¹ one can compute the last row of B¹ .

Ask rest question again with proper image as some words are not clear.