Tagged: determinant of a matrix

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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Linear Algebra Midterm 1 at the Ohio State University (2/3)

Problem 571

The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.
There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).
The time limit was 55 minutes.


This post is Part 2 and contains Problem 4, 5, and 6.
Check out Part 1 and Part 3 for the rest of the exam problems.


Problem 4. Let
\[\mathbf{a}_1=\begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix}, \mathbf{a}_2=\begin{bmatrix}
2 \\
-1 \\
4
\end{bmatrix}, \mathbf{b}=\begin{bmatrix}
0 \\
a \\
2
\end{bmatrix}.\]

Find all the values for $a$ so that the vector $\mathbf{b}$ is a linear combination of vectors $\mathbf{a}_1$ and $\mathbf{a}_2$.


Problem 5.
Find the inverse matrix of
\[A=\begin{bmatrix}
0 & 0 & 2 & 0 \\
0 &1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1
\end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason.


Problem 6.
Consider the system of linear equations
\begin{align*}
3x_1+2x_2&=1\\
5x_1+3x_2&=2.
\end{align*}

(a) Find the coefficient matrix $A$ of the system.

(b) Find the inverse matrix of the coefficient matrix $A$.

(c) Using the inverse matrix of $A$, find the solution of the system.

(Linear Algebra Midterm Exam 1, the Ohio State University)
 

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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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How to Prove a Matrix is Nonsingular in 10 Seconds

Problem 509

Using the numbers appearing in
\[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\] we construct the matrix \[A=\begin{bmatrix}
3 & 14 &1592& 65358\\
97932& 38462643& 38& 32\\
7950& 2& 8841& 9716\\
939937510& 5820& 974& 9
\end{bmatrix}.\]

Prove that the matrix $A$ is nonsingular.

 

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The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$

Problem 505

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.
Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
\[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\]

Using the formula, calculate the inverse matrix of $\begin{bmatrix}
2 & 1\\
1& 2
\end{bmatrix}$.

 

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Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

Problem 486

Determine whether there exists a nonsingular matrix $A$ if
\[A^4=ABA^2+2A^3,\] where $B$ is the following matrix.
\[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
2 & 1 & -4
\end{bmatrix}.\]

If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

(The Ohio State University, Linear Algebra Final Exam Problem)
 

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Eigenvalues of Orthogonal Matrices Have Length 1. Every $3\times 3$ Orthogonal Matrix Has 1 as an Eigenvalue

Problem 419

(a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$


(b) Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.

 

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Determine Whether Given Matrices are Similar

Problem 391

(a) Is the matrix $A=\begin{bmatrix}
1 & 2\\
0& 3
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
3 & 0\\
1& 2
\end{bmatrix}$?  

(b) Is the matrix $A=\begin{bmatrix}
0 & 1\\
5& 3
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
1 & 2\\
4& 3
\end{bmatrix}$? 

(c) Is the matrix $A=\begin{bmatrix}
-1 & 6\\
-2& 6
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
3 & 0\\
0& 2
\end{bmatrix}$? 

(d) Is the matrix $A=\begin{bmatrix}
-1 & 6\\
-2& 6
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
1 & 2\\
-1& 4
\end{bmatrix}$?

 

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Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)

Problem 389

(a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.
Find $\det(A)$.

(b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.

(c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of $A$?

(Harvard University, Linear Algebra exam problem)

 

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Determinant of a General Circulant Matrix

Problem 374

Let \[A=\begin{bmatrix}
a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\
a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\
a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\
\vdots & \vdots & \dots & \vdots & \vdots \\
a_{2} & a_3 & \dots & a_{0} & a_{1}\\
a_{1} & a_2 & \dots & a_{n-1} & a_{0}
\end{bmatrix}\] be a complex $n \times n$ matrix.
Such a matrix is called circulant matrix.
Then prove that the determinant of the circulant matrix $A$ is given by
\[\det(A)=\prod_{k=0}^{n-1}(a_0+a_1\zeta^k+a_2 \zeta^{2k}+\cdots+a_{n-1}\zeta^{k(n-1)}),\] where $\zeta=e^{2 \pi i/n}$ is a primitive $n$-th root of unity.

 

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Quiz 11. Find Eigenvalues and Eigenvectors/ Properties of Determinants

Problem 363

(a) Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
3 & -2\\
6& -4
\end{bmatrix}.\]

(b) Let
\[A=\begin{bmatrix}
1 & 0 & 3 \\
4 &5 &6 \\
7 & 0 & 9
\end{bmatrix} \text{ and } B=\begin{bmatrix}
2 & 0 & 0 \\
0 & 3 &0 \\
0 & 0 & 4
\end{bmatrix}.\] Then find the value of
\[\det(A^2B^{-1}A^{-2}B^2).\] (For part (b) without computation, you may assume that $A$ and $B$ are invertible matrices.)

 

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