031 Review Problems

This is a list of sample problems and is meant to represent the material usually covered in Math 31. An actual test may or may not be similar.

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

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

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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.

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

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

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

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

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

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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 $B=C.$

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10.

(a) Is the matrix $A={\begin{bmatrix}3&1\\0&3\end{bmatrix}}$ diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

(b) Is the matrix $A={\begin{bmatrix}2&0&-2\\1&3&2\\0&0&3\end{bmatrix}}$ diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

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