Which of the following statements are true? A. The equation Ax = b has a unique

Question

Which of the following statements are true? A. The equation Ax = b has a unique solution if and only if the matrix A is singular. If the equation Ax = b does not have a solution, then the matrix A is singular. The matrix .1 is singular if the augmented matrix-1 has a pivot position in every row. If A is a singular n × n matrix, then A cannot have a pivot position in every row. Any linear combination of n-vectors in R" can always be written in the form Ar for a suitable matrix A and vector An n-vector b is a linear combination of the columns of an n x n matrix A if and only if the equation Ar b has at least one solution B. C. D. E. F.

Explanation / Answer

A. False statement. The equation Ax = b has a unique solution if and only if the matrix A is non-singular.

B. True statement. If the matrix A is non-singular, then x = A-1b is the solution.

C. False statement. The system Ax = b is inconsistent only if [A b] has a pivot in the last ("b") column and not otherwise. Thus A is singular only if [A b] has a pivot in the last ("b") column.

D. False statement . If A is a singular nxn matrix, then the equation AX = b will not have a unique solution, but it can have infinite solutions in which case, A can have a pivot in every row.

E. True statement.

F. True statement.

Related Questions

Navigate

• Browse (All)
• Browse W
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.