Now we calculate the two determinants in the second equality separately.
The easiest way is to note that the determinant of a triangular matrix is the product of its diagonal entries.
Thus the first determinant is $(-\lambda)^{n-1}$ and the second determinant is $1$.
(If you don’t know this fact, then use the first row cofactor expansion inductively to compute the first determinant. For the second one, use the first column cofactor expansion inductively.)
Thus we obtain $\det(A-\lambda I)=(-1)^n\lambda^n+(-1)^{n+1}=(-1)^n(\lambda^n-1)$.
Therefore eigenvalues are $n$-th roots of unity $e^{2\pi i/n}$ for $i=0,1,\dots, n-1$.
Comment.
The original determinant is not in a good shape for induction but once we apply the 1st row cofactor expansion the smaller determinants obtained are better suited for induction.
When I say in the proof “inductively”, I meant that you need to use mathematical induction to prove the claim (more) rigorously.
Rotation Matrix in Space and its Determinant and Eigenvalues
For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\[A=\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
\end{bmatrix}.\]
(a) Find the determinant of the matrix $A$.
(b) Show that $A$ is an […]
Maximize the Dimension of the Null Space of $A-aI$
Let
\[ A=\begin{bmatrix}
5 & 2 & -1 \\
2 &2 &2 \\
-1 & 2 & 5
\end{bmatrix}.\]
Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.
Your score of this problem is equal to that […]
Compute Determinant of a Matrix Using Linearly Independent Vectors
Let $A$ be a $3 \times 3$ matrix.
Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have
\[A\mathbf{x}=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, A\mathbf{y}=\begin{bmatrix}
0 \\
1 \\
0
[…]
Find All the Eigenvalues of Power of Matrix and Inverse Matrix
Let
\[A=\begin{bmatrix}
3 & -12 & 4 \\
-1 &0 &-2 \\
-1 & 5 & -1
\end{bmatrix}.\]
Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.
Proof.
We first determine all the eigenvalues of the matrix […]
If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix
Let $A, B$ be complex $2\times 2$ matrices satisfying the relation
\[A=AB-BA.\]
Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.
Hint.
Find the trace of $A$.
Use the Cayley-Hamilton theorem
Proof.
We first calculate the […]
How to Find the Determinant of the $3\times 3$ Matrix
Find the determinant of the matix
\[A=\begin{bmatrix}
100 & 101 & 102 \\
101 &102 &103 \\
102 & 103 & 104
\end{bmatrix}.\]
Solution.
Note that the determinant does not change if the $i$-th row is added by a scalar multiple of the $j$-th row if $i \neq […]