2.3 Exercises

20 points · 4 extra credit

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Characterizations of Invertible Matrices

1. (3 pts) Determine whether

$A = \begin{bmatrix} 1 & 0 & -2 \\ 3 & 1 & -2 \\ -5 & -1 & 9 \end{bmatrix}$

is invertible by row reducing it. Count the pivot positions and state which statement of the Invertible Matrix Theorem you used to reach your conclusion.
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2. (3 pts) Find all values of $h$ for which

$A = \begin{bmatrix} 1 & -2 & 1 \\ 2 & -3 & h \\ 1 & 1 & 1 \end{bmatrix}$

is not invertible.
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3. (3 pts) Let $A$ be an $n\times n$ matrix, and suppose the equation $A\mathbf{x} = 0$ has only the trivial solution. Using the Invertible Matrix Theorem, explain why $A\mathbf{x} = \mathbf{b}$ is consistent for every $\mathbf{b} \in \mathbb{R}^n$ and has exactly one solution.
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4. (3 pts) State whether each of the following is true or false, and justify each using the Invertible Matrix Theorem. Assume $A$ is $n\times n$ .

(a) If $A$ has $n$ pivot positions, then $A$ is invertible.

(b) If the columns of $A$ span $\mathbb{R}^n$ , then $A$ might still fail to be invertible.

(c) If $A$ is invertible, then its reduced echelon form is $I_n$ .

(d) If there is an $n\times n$ matrix $C$ with $CA = I$ , then $A$ is invertible.
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5. (2 pts) Suppose $A$ is a $5\times5$ matrix for which the equation $A\mathbf{x} = 0$ has a nontrivial solution. List three statements from the Invertible Matrix Theorem that must therefore be false for $A$ .
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6. (2 pts) Explain why a square matrix with two identical rows cannot be invertible. Reference an appropriate statement of the Invertible Matrix Theorem.
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7. (4 pts) Let $A$ and $B$ be $n\times n$ matrices with $AB = I$ . Prove that $B$ is invertible with $B^{-1} = A$ , and hence that $A$ is invertible as well. (Do not assume $BA = I$ ; deduce it.)
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8. (4 pts ••) A square matrix $A$ is called nilpotent if $A^k = 0$ for some integer $k \ge 1$ . Prove that if $A$ is nilpotent, then $I - A$ is invertible, and

$(I - A)^{-1} = I + A + A^2 + \cdots + A^{k-1}.$
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