The Timpson Lyceum · Linear Algebra (2026)

2.3 Exercises

Problem sheet · 20 points · 4 extra credit

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.
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.
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.
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.
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 $A$ .
6. (2 pts) Explain why a square matrix with two identical rows cannot be invertible. Reference an appropriate statement of the Invertible Matrix Theorem.
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.)
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

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