The given matrix is an augmented matrix representing a system of linear equations in x, y, and z. Use the Gauss-Jordan elimination method (see Gauss-Jordan elimination method box and Example 1) to find the solution of the system. ſi 2 51 | 2 - 4 LO 1 - 3 (x, y, z) =(
1) Consider the system of linear algebraic equations Ax = B where | 1 1/2 1/31 1/2 1/3 1/4 11/3 1/4 1/5 a) Find x, A" and det(A) using Gauss-Jordan elimination without pivoting. b) Using the result of part (a), find the condition number of A based on the Euclidean (Frobenius) norm. How many digits of precision do you suspect are lost in the solution x due to ill-conditioning?
4. (10 pts) Using the Gauss-Jordan elimination process, solve the following systems of linear equations. How many solutions are there? Can we apply Cramer's rule? Explain why (Use the matrix form of linear equations.) 4. (10 pts) Using the Gauss-Jordan elimination process, solve the following systems of linear equations. How many solutions are there? Can we apply Cramer's rule? Explain why (Use the matrix form of linear equations.)
Problem 3 1. Prove that B (51, b2, b3,-4} {а, ег#3+ега, +6) is the basis for R4. al 2. Find 1 4 0 0 0 : 0 0 0 00 0 b 3. Consider the map T: R4-W with B-matrix B a 。), Find the standard matrix 1896 of T Problem 3 1. Prove that B (51, b2, b3,-4} {а, ег#3+ега, +6) is the basis for R4. al 2. Find 1 4 0 0 0 : 0 0 0 00...
4 Let A12 and b4 14 (a) Find A-1 and use it solve the four equations Ax-b1, Ax b2 Ax b3, and Ax b4 (b) The four equations in part (a) can be solved by the same set of operations, since the coefficient matrix is the same in each case Solve the four equations in part (a) by row reducing the augmented matrix [A bj b2 b3 b4
Consider the following. (x1 - x2 + 4x3 = 20 3x + 332 = -4 -6x2 + 5x3 = 32 (a) Write the system of linear equations as a matrix equation, AX = B. 14 X1 I X2 = IL X3] (b) Use Gauss-Jordan elimination on [ A B] to solve for the matrix X. X2
2,3, 6, 7 1. Without matrices, solve the following system using the Gaussian elimination method + 1 + HP 6x - Sy- -2 2. Consider the following linear system of equation 3x 2 Sy- (a) Write the augmented matrix for this linear system (b) Use row operations to transform the augmented matrix into row.echelon form (label all steps) (c) Use back substitution to solve the linear system. (find x and y) x + 2y 2x = 5 3. Consider the...
1. For each of the following systems of linear equations, find: • the augmented matrix • the coefficient matrix • the reduced row echelon form of the augmented matrix • the rank of the augmented matrix • all solutions to the original system of equations Show your work, and use Gauss-Jordan elimination (row reduction) when finding the reduced row echelon forms. (b) 2 + 2x W 2w - 2y - y + y + 3z = 0 = 1 +...
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without partial pivoting. (10 marks) 7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without...
Find the general solution for the augmented matrices 1-2 (1 1 3 2 -1 -1 4 1 ) -2) Solve the system (shown here as an augmented matrix) by Gauss-Jordan elimination