Consider the system of lienar equations: ar +by = c dr + ey=f (a) Write the...
Determine the values of a for which the following
system of linear equations has no solutions, a unique solution, or
infinitely many solutions. You can select 'always', 'never', 'a =
', or 'a ≠', then specify a value or comma-separated list of
values.
x1+ax2−x3
=
2
−x1+4x2−2x3
=
−5
−2x1+3x2+x3
=
−4
No Solutions:
Unique Solution:
Infinitely Many Solutions:
Describe the solutions of the first system of equations below in parametric vector form. Provide a geometric comparison with the solution set of the second system of equations below. 4x1 +4x2+8X3 = 16 - 12X1 - 12X2 - 24x3 = - 48 - 6x2 - 6x3 = 18 4x7 +4x2+8X3 = 0 - 12X1 - 12X2 - 24x3 = 0 - 6x2 - 6x3 = 0 X1 Describe the solution set, x = X2 of the first system of equations...
2. Given that u., and ware three solutions of the linear system Az = b. Verify that the vector cu+du+ (1-c-d)w is also a solution of Ar = b for any scalars DER - 2 1 1 Let A = 1 1 - 2 1 Determine whether the system Az = b is consistent for every beR. 1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only...
3. Consider the following system of linear equations: 2.0 + 2y + 2kz = 2 kx + ky+z=1 2x + 3y + 72 = 4 (i) Turn the system into row echelon form. (ii) Determine which values of k give (i) a unique solution (ii) infinitely many solutions and (iii) no solutions. Show your working. 4. Solve the following system of linear equations using Gauss-Jordan elimination: x1 + x2 - 2.13 + 24 +3.25 = 1 2.x1 - x2 +...
Solve the system of equations using substitutions
Solve the system of equations using substitution. y=--x + 2 2 (1) y+2x = 5 Select the correct choice below and, if necessary,fill in the answer box to complete your choice. 0 A. The solution is O B. ° C. The solution is the empty set. (Type an ordered pair. Type an integer or a simplified fraction.) There are infinitely many solutions of the form (x.yl (Type an equation.)
UNULUI UU Two systems of equations are given below. For each system, choose the best description c If applicable, give the solution. The syst The syst x - 3y = -9 -x + 3y = 9 (x, y) = The syst They mu The syst The syst X - 31 = 3 -X - 3 = 3 The syst They m Two systems of equations are given below. For each system, choose the best description of its solution. If applicable,...
1. Consider the following augmented matrix of a system of linear equations: [1 1 -2 2 3 1 2 -2 2 3 0 0 1 -1 3 . The system has 0 0 -1 2 -3 a) a unique solution b) no solutions c) infinitely many solutions with one free variable d) infinitely many solutions with two variables e) infinitely many solutions with three variables
L. Answer True or False. Justify your answer (a) Every linear system consisting of 2 equations in 3 unknowns has infinitely many solutions (b) If A. B are n × n nonsingular matrices and AB BA, then (e) If A is an n x n matrix, with ( +A) I-A, then A O (d) If A, B two 2 x 2 symmetric matrices, then AB is also symmetric. (e) If A. B are any square matrices, then (A+ B)(A-B)-A2-B2 2....
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions. If A is square and invertible, there is a unique solution, which can be written as x = A-'b. The concept of pseudoinverse seeks to generalise this idea to non-square matrices and to cases (ii) and (iii). Taking case (ii) of an inconsistent linear system, we may solve the normal equations AT Ar = Ab...
Write each statement as True or False (a) If an (nx n) matrix A is not invertible then the linear system Ax-O hns infinitely many b) If the number of equations in a linear system exceeds the number of unknowns then the system 10p solutions must be inconsistent ) If each equation in a consistent system is multiplied through by a constant c then all solutions to the new system can be obtained by multiplying the solutions to the original...