Series Problems

These questions are meant to be additional practice problems for series.

Determine whether the series converge or diverge.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

True or false: If all the entries of a  $7\times 7$  matrix  $A$  are  $7,$  then  ${\text{det }}A$  must be  $7^{7}.$

 Problem 2 

True or false: If a matrix  $A^{2}$  is diagonalizable, then the matrix  $A$  must be diagonalizable as well.

 Problem 3 

True or false: If  $A$  is a  $4\times 4$  matrix with characteristic equation  $\lambda (\lambda -1)(\lambda +1)(\lambda +e)=0,$  then  $A$  is diagonalizable.

 Problem 4 

True or false: If  $A$  is invertible, then  $A$  is diagonalizable.

 Problem 5 

True or false: If  $A$  and  $B$  are invertible  $n\times n$  matrices, then so is  $A+B.$

 Problem 6 

True or false: If  $A$  is a  $3\times 5$  matrix and  ${\text{dim Nul }}A=2,$  then  $A{\vec {x}}={\vec {b}}$  is consistent for all  ${\vec {b}}$  in  $\mathbb {R} ^{3}.$

 Problem 7 

True or false: Let  $C=AB$  for  $4\times 4$  matrices  $A$  and  $B.$  If  $C$  is invertible, then  $A$  is invertible.

 Problem 8 

True or false: Let  $W$  be a subspace of  $\mathbb {R} ^{4}$  and  ${\vec {v}}$  be a vector in  $\mathbb {R} ^{4}.$  If  ${\vec {v}}\in W$  and  ${\vec {v}}\in W^{\perp },$  then  ${\vec {v}}={\vec {0}}.$

 Problem 9 

True or false: If  $A$  is an invertible  $3\times 3$  matrix, and  $B$  and  $C$  are  $3\times 3$  matrices such that  $AB=AC,$  then  $B=C.$