Suppose that $A$ is an $n\times n$ singular matrix.
Prove that for sufficiently small $\epsilon>0$, the matrix $A-\epsilon I$ is nonsingular, where $I$ is the $n \times n$ identity matrix.

Consider the characteristic polynomial $p(t)$ of the matrix $A$.

Note that the eigenvalues of $A$ are the roots of $p(t)$.

Thus if $\epsilon$ is not an eigenvalue, then $p(\epsilon)\neq 0$.

Proof.

Let $p(t)=\prod_{i=}^n(\lambda_i-t)$ be the characteristic polynomial for $A$, where $\lambda_i$ are eigenvalues of $A$.

Let $\lambda_{i_0}$ be the nonzero eigenvalue of $A$ of the smallest absolute value.
That is $|\lambda_{i_0}|\leq |\lambda_i|$ for any nonzero eigenvalue $\lambda_i$.
Then for any $0< \epsilon <|\lambda_{i_0}|$, we have $\det(A-\epsilon I)=p(\epsilon)\neq 0$, otherwise $\epsilon$ would be an eigenvalue of $A$ but it is impossible because of the minimality of $\lambda_{i_0}$.

Therefore the matrix $A-\epsilon I$ is nonsingular for all $0<\epsilon<|\lambda_{i_0}|$.

Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$
Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.
What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?
(The Ohio State University, Linear Algebra Final Exam […]

Find the Rank of the Matrix $A+I$ if Eigenvalues of $A$ are $1, 2, 3, 4, 5$
Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.
(UCB-University of California, Berkeley, […]

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 […]

Find All Values of $x$ so that a Matrix is Singular
Let
\[A=\begin{bmatrix}
1 & -x & 0 & 0 \\
0 &1 & -x & 0 \\
0 & 0 & 1 & -x \\
0 & 1 & 0 & -1
\end{bmatrix}\]
be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.
Hint.
Use the fact that a matrix is singular if and only […]

Diagonalize a 2 by 2 Symmetric Matrix
Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
Solution.
The characteristic polynomial $p(t)$ of the matrix $A$ […]

Subspaces of Symmetric, Skew-Symmetric Matrices
Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.
(a) The set $S$ consisting of all $n\times n$ symmetric matrices.
(b) The set $T$ consisting of […]

Characteristic Polynomials of $AB$ and $BA$ are the Same
Let $A$ and $B$ be $n \times n$ matrices.
Prove that the characteristic polynomials for the matrices $AB$ and $BA$ are the same.
Hint.
Consider the case when the matrix $A$ is invertible.
Even if $A$ is not invertible, note that $A-\epsilon I$ is invertible matrix […]

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
[…]

[…] note that $A-epsilon I$ is invertible matrix for sufficiently small $epsilon$. (See Problem Perturbation of a singular matrix is nonsingular for a proof of this […]

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[…] note that $A-epsilon I$ is invertible matrix for sufficiently small $epsilon$. (See Problem Perturbation of a singular matrix is nonsingular for a proof of this […]