Question-74

Let \(\displaystyle A\) be a \(\displaystyle n\times n\) real matrix. Which of the following statements are equivalent to: “the matrix \(\displaystyle A\) is invertible”?

• The columns of \(\displaystyle A\) are linearly independent.

• The columns of \(\displaystyle A\) span \(\displaystyle \mathbb{R}^{n}\).

• The rows of \(\displaystyle A\) are linearly independent.

• The rank of \(\displaystyle A\) is \(\displaystyle n\).

• The nullity of \(\displaystyle A\) is zero.

• The only solution of the system \(\displaystyle Ax=0\) is \(\displaystyle x=0\).

• The function \(\displaystyle T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) given by \(\displaystyle T( x) =Ax\) is bijective

• \(\displaystyle \text{det}( A) \neq 0\)

• \(\displaystyle A\) doesn’t have a zero eigenvalue.

• \(\displaystyle A\) doesn’t have a zero singular value.

Answer

• The columns of \(\displaystyle A\) are linearly independent.

• The columns of \(\displaystyle A\) span \(\displaystyle \mathbb{R}^{n}\).

• The rows of \(\displaystyle A\) are linearly independent.

• The rank of \(\displaystyle A\) is \(\displaystyle n\).

• The nullity of \(\displaystyle A\) is zero.

• The only solution of the system \(\displaystyle Ax=0\) is \(\displaystyle x=0\).

• The function \(\displaystyle T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) given by \(\displaystyle T( x) =Ax\) is bijective

• \(\displaystyle \text{det}( A) \neq 0\)

• \(\displaystyle A\) doesn’t have a zero eigenvalue.

• \(\displaystyle A\) doesn’t have a zero singular value.

Solution

All of them are true! The proof for each one follows from the theory and is therefore left out here.