Question

Q3 (8 points) In the following A is a 3 × 4 matrix (3 rows, 4 columns) and the coefficient matrix of a system of linear equations. A. Find an example of such a matrix A and a vector b such that the system with augmented matrix [A | b] has no solution. Justify your answer. B. Find an example of such a matrix A and a vector b such that the system with augmented matrix [A | b] has infinitely many solutions. Justify your answer. C. Is there an example A, b such that the system with augmented matrix [A | b] has a unique solution? If so, give one, if not, why not? D. Give an example of a coefficient matrix A such that [A | b] has a solution for every vector b in R 3

Answer 1

A. Let A =

1	0	1	2
2	1	3	5
0	0	0	0

and b = (1,2,3)^T. It may be observed that b cannot be expressed as a linear combination of the columns of A. Hence the linear system with the augmented matrix [ A|b] has no solution.

B. Let A be as above and let b = (1,3,0)^T. Then b is the sum of the first 2 columns of A. Hence the linear system with the augmented matrix [ A|b] is consistent. Also, since A is not a square matrix, hence this system has infinite solutions ( A consistent linear system has either a unique solution or infinite solutions).

C. If A is a 3x4 matrix, then there cannot be a linear system with augmented matrix [A|b] with only one solution. Either there will be infinite solutions or no solution.

There cwill be a unique solution only when A is a square matrix which is invertible.

D. When the columns of A span R³, then every b in R³ will be a linear combination of the columns of A. Hence the linear system with the augmented matrix [ A|b] will be consistent for every vector b in R³. For example, let A =

1	0	0	1
0	1	0	1
0	0	1	1

Then the columns of A span R³ so that every b in R³ will be a linear combination of the columns of A and hence the linear system with the augmented matrix [ A|b] will be consistent for every vector b in R³.