Solution Found!
Suppose A is n × n and the equation Ax = 0 has only the
Chapter 2, Problem 23E(choose chapter or problem)
Problem 23E
Suppose A is n × n and the equation Ax = 0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to In. By Theorem 7, this shows that A must be invertible. (This exercise and Exercise 24 will be cited in Section 2.3.)
Theorem 7:
An n × n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A–1.
Questions & Answers
QUESTION:
Problem 23E
Suppose A is n × n and the equation Ax = 0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to In. By Theorem 7, this shows that A must be invertible. (This exercise and Exercise 24 will be cited in Section 2.3.)
Theorem 7:
An n × n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A–1.
ANSWER:
Solution 23E