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A matrix A is invertible if and only if

• its determinant is non-zero

• there exists a matrix B such that AB=I

• there is a matrix C such that CA=I

• it is one to one as a linear transformation

It is possible for

 $A=A^T$  

• true

• false

the sum of the diagonal entries of a matrix is its

The augmented matrix

 $\left[A\ b\right]$  is shorthand for which matrix equation?

•  $\left[A\right]\cdot\overrightarrow{x}=\overrightarrow{0}$  

•  $\left[A\right]\cdot\overrightarrow{x}=\overrightarrow{b}$  

•  $\left[Ax\right]=b$  

•  $\left[A0\right]=b$  

If

 $A\in M_{n\times m}\left(\R\right)$  with  $n\ne m$  and  $p\left(x\right)$  is a  polynomial, then  $p\left(A\right)$  is well defined.

• True

• False

Which of the following is not a vector space

• 
  

  $M_{n\times m}\left(\mathbb{C}\right)$
  

• the set of polynomials of degree exactly 2

• $\mathbb{R}^n$

• the set of all polynomials

It is possible to have a vector space over  $\R$  with three elements. 

• True

• False

If U and W are subspaces of a vector space V, then which of the following is not a subspace of V

• $U\cup W$

• $U\cap W$

• $U+W$

• $U\oplus W$

Every vector space (including infinite dimensional ones) has a basis

• True

• False

If  $A\in M_n\left(\mathbb{R}\right)$  , then  $A^{\ast}=A^T$ 

• True

• False