Tagged: invertible matrix

Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix

Problem 583

Consider the $2\times 2$ complex matrix
\[A=\begin{bmatrix}
a & b-a\\
0& b
\end{bmatrix}.\]

(a) Find the eigenvalues of $A$.

(b) For each eigenvalue of $A$, determine the eigenvectors.

(c) Diagonalize the matrix $A$.

(d) Using the result of the diagonalization, compute and simplify $A^k$ for each positive integer $k$.

 
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Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible?

Problem 582

A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix.
Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.

Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of the same size as $A$.
Is the matrix $B-A$ invertible? If so prove it. Otherwise, give a counterexample.

 
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Two Matrices are Nonsingular if and only if the Product is Nonsingular

Problem 562

An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.
Using the definition of a nonsingular matrix, prove the following statements.

(a) If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular.

(b) Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then:

  1. The matrix $B$ is nonsingular.
  2. The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.)

 
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Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations

Problem 552

For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.

(a) $A=\begin{bmatrix}
1 & 3 & -2 \\
2 &3 &0 \\
0 & 1 & 1
\end{bmatrix}$
 
(b) $A=\begin{bmatrix}
1 & 0 & 2 \\
-1 &-3 &2 \\
3 & 6 & -2
\end{bmatrix}$.

 
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A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$

Problem 548

An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that

  1. $AB=I$, and
  2. $BA=I$,

where $I$ is the $n\times n$ identity matrix.

If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted by $A^{-1}$.


In this problem, we prove that if $B$ satisfies the first condition, then it automatically satisfies the second condition.
So if we know $AB=I$, then we can conclude that $B=A^{-1}$.


Let $A$ and $B$ be $n\times n$ matrices.
Suppose that we have $AB=I$, where $I$ is the $n \times n$ identity matrix.

Prove that $BA=I$, and hence $A^{-1}=B$.

 
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Find Inverse Matrices Using Adjoint Matrices

Problem 546

Let $A$ be an $n\times n$ matrix.

The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be
\[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column.

Then consider the $n\times n$ matrix $C=(C_{ij})$, and define the $n\times n$ matrix $\Adj(A)=C^{\trans}$.
The matrix $\Adj(A)$ is called the adjoint matrix of $A$.

When $A$ is invertible, then its inverse can be obtained by the formula

\[A^{-1}=\frac{1}{\det(A)}\Adj(A).\]

For each of the following matrices, determine whether it is invertible, and if so, then find the invertible matrix using the above formula.

(a) $A=\begin{bmatrix}
1 & 5 & 2 \\
0 &-1 &2 \\
0 & 0 & 1
\end{bmatrix}$.

 
(b) $B=\begin{bmatrix}
1 & 0 & 2 \\
0 &1 &4 \\
3 & 0 & 1
\end{bmatrix}$.

 
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10 True of False Problems about Nonsingular / Invertible Matrices

Problem 500

10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.

The quiz is designed to test your understanding of the basic properties of these topics.

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.

 
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Eigenvalues of Similarity Transformations

Problem 452

Let $A$ be an $n\times n$ complex matrix.
Let $S$ be an invertible matrix.

(a) If $SAS^{-1}=\lambda A$ for some complex number $\lambda$, then prove that either $\lambda^n=1$ or $A$ is a singular matrix.

(b) If $n$ is odd and $SAS^{-1}=-A$, then prove that $0$ is an eigenvalue of $A$.

(c) Suppose that all the eigenvalues of $A$ are integers and $\det(A) > 0$. If $n$ is odd and $SAS^{-1}=A^{-1}$, then prove that $1$ is an eigenvalue of $A$.

 
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True of False Problems on Determinants and Invertible Matrices

Problem 438

Determine whether each of the following statements is True or False.

(a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.

(b) If the characteristic polynomial of an $n \times n$ matrix $A$ is
\[p(\lambda)=(\lambda-1)^n+2,\] then $A$ is invertible.

(c) If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.

(d) If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

(e) If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.

(Stanford University, Linear Algebra Exam Problem)
 
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If a Matrix is the Product of Two Matrices, is it Invertible?

Problem 393

(a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as
\[A=BC,\] where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.

Prove that the matrix $A$ cannot be invertible.


(b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be written as
\[A=BC,\] where $B$ is a $ 2\times 3$ matrix and $C$ is a $3\times 2$ matrix.

Can the matrix $A$ be invertible?

 
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