031 Review Part 1

These questions are from sample exams and actual exams at other universities. The questions are meant to represent the material usually covered in Math 31 for the final. An actual test may or may not be similar.

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 ${\vec {v}}={\vec {0}}.$