Let $p(t)$ be the characteristic polynomial of the matrix $A$.
Then we have
\[p(t)=\det(A-tI)=\prod_{i=1}^n(\lambda_i-t)=(-1)^n\prod_{i=1}^n(t-\lambda_i),\]
where $\lambda_i$ are eigenvalues of $A$.
Thus substituting $t=1$, we have
\[\det(I-A)=(-1)^n(-1)^n\prod_{i=1}^n(1-\lambda_i)=\prod_{i=1}^n(1-\lambda_i)>0.\]
Since eigenvalues $\lambda_i$ are less than $1$, the product is positive, and this proves $\det(I-A)>0$.
Solution 2.
There exists an invertible matrix $P$ such that the matrix $P^{-1}AP$ is the Jordan canonical form. That is $P^{-1}AP$ is an upper triangular matrix whose diagonal entries are eigenvalues $\lambda_i$ of the matrix $A$.
Then we have
\begin{align*}
\det(I-A)=\det(I-P^{-1}AP)=\prod_{i=1}^n(1-\lambda_i).
\end{align*}
Here the last equality follows from the fact that the determinant of an upper triangular matrix is the product of its diagonal entries.
Since the eigenvalues $\lambda_i$ are less than $1$, the last product is positive. Thus we proved $\det(I-A)>0$.
Determinant/Trace and Eigenvalues of a Matrix
Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.
Show that
(1) $$\det(A)=\prod_{i=1}^n \lambda_i$$
(2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$
Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix […]
Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$
Let $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.)
For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of […]
If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent
Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$.
Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix.
Steps.
Use the Jordan canonical form of the matrix $A$.
We want […]
A Square Root Matrix of a Symmetric Matrix
Answer the following two questions with justification.
(a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix.
(b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]
Finite Order Matrix and its Trace
Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that
(a) $|\tr(A)|\leq n$.
(b) If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$.
(c) $\tr(A)=n$ if and only if $A=I_n$.
Proof.
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Nilpotent Matrix and Eigenvalues of the Matrix
An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix.
Prove the followings.
(a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.
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Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?
Find a square root of the matrix
\[A=\begin{bmatrix}
1 & 3 & -3 \\
0 &4 &5 \\
0 & 0 & 9
\end{bmatrix}.\]
How many square roots does this matrix have?
(University of California, Berkeley Qualifying Exam)
Proof.
We will find all matrices $B$ such that […]
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Let $T=\begin{bmatrix}
1 & 0 & 2 \\
0 &1 &1 \\
0 & 0 & 2
\end{bmatrix}$.
Calculate and simplify the expression
\[-T^3+4T^2+5T-2I,\]
where $I$ is the $3\times 3$ identity matrix.
(The Ohio State University Linear Algebra Exam)
Hint.
Use the […]