Question

For this exercise assume that the matrices are all n x n. Each part of this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever

"statement 1" happens to be true. An implication is False if there is an instance in which "statement 2" is false but
"statement 1" is true. Justify each answer.

If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n × n identity matrix

If the columns of A span ℝ^n , then the columns are linearly independent

If A is an n × n matrix, then the equation Ax = b has at least one solution for each b in ℝ^n

Answer 1

True

True

False

Explanation:

TRUE

If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n × n identity matrix

Here's why

If the equation Ax = 0 has only the trivial solution the determinant of the matrix is NOT 0 and the matrix is invertible therefore it is row equivalent to the nxn identity matrix.

TRUE

If the columns of A span ℝ^n , then the columns are linearly independent

Here's why

Remember that the rank nullity theorem states that

`\text{rank}(A) + \text{Nullity}(A) = \text{Dim}(V)`

According to the information given we know that

`\text{rank}(A) = n \\dim(V) = n \\`

Therefore you have

`n + \text{Nullity}(A) = n`

and

`\text{Nullity}(A) = 0`

Which is equivalent to the problem we just solved.

FALSE

If A is an n × n matrix, then the equation Ax = b has at least one solution for each b in ℝ^n

Here's why

Take b as a non null vector and A=0, then Ax = 0 and Ax=b will have no solution.