Question

Let \(A\) be an \(n \times n\) matrix of real or complex numbers. Which of the following statements are equivalent to: "the matrix \(A\) is invertible"?

a) The columns of \(A\) are linearly independent.
b) The columns of \(A \operatorname{span} \mathbb{R}^{n}\).
c) The rows of \(A\) are linearly independent.
d) The kernel of \(A\) is 0 .
e) The only solution of the homogeneous equations \(A x=0\) is \(x=0\).
f) The linear transformation \(T_{A}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) defined by \(A\) is 1-1.
g) The linear transformation \(T_{A}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) defined by \(A\) is onto.
h) The rank of \(A\) is \(n\).
i) The adjoint, \(A^{*}\), is invertible.
j) \(\operatorname{det} A \neq 0\).