Tagged: determinant

Special Linear Group is a Normal Subgroup of General Linear Group

Problem 332

Let $G=\GL(n, \R)$ be the general linear group of degree $n$, that is, the group of all $n\times n$ invertible matrices.
Consider the subset of $G$ defined by
\[\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.\] Prove that $\SL(n, \R)$ is a subgroup of $G$. Furthermore, prove that $\SL(n,\R)$ is a normal subgroup of $G$.
The subgroup $\SL(n,\R)$ is called special linear group

 
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Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation

Problem 330

Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
\[ T(X)=AX-XA\] for each $X\in V$.

(a) Prove that $T:V\to V$ is a linear transformation.

(b) Let $B$ be a basis of $V$. Let $P$ be the matrix representation of $T$ with respect to $B$. Find the determinant of $P$.

 
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Is there an Odd Matrix Whose Square is $-I$?

Problem 316

Let $n$ be an odd positive integer.
Determine whether there exists an $n \times n$ real matrix $A$ such that
\[A^2+I=O,\] where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix.

If such a matrix $A$ exists, find an example. If not, prove that there is no such $A$.

How about when $n$ is an even positive number?

 
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Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation

Problem 289

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

(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\] where
\[B=\begin{bmatrix}
4 & 1\\
2& 6
\end{bmatrix}.\] Verify that the matrix $A$ you obtained is actually a nonsingular matrix.

(The Ohio State University, Linear Algebra Midterm Exam Problem)
 
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Idempotent Matrices. 2007 University of Tokyo Entrance Exam Problem

Problem 265

For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions.

  1. $A=aP+(a+1)Q$
  2. $P^2=P$
  3. $Q^2=Q$
  4. $PQ=O$
  5. $QP=O$,

where $O$ is the $2\times 2$ zero matrix.
Then do the following problems.


(a) Prove that $(P+Q)A=A$.


(b) Suppose $a$ is a positive real number and let
\[ A=\begin{bmatrix}
a & 0\\
1& a+1
\end{bmatrix}.\] Then find all matrices $P, Q$ satisfying conditions (1)-(5).


(c) Let $n$ be an integer greater than $1$. For any integer $k$, $2\leq k \leq n$, we define the matrix
\[A_k=\begin{bmatrix}
k & 0\\
1& k+1
\end{bmatrix}.\] Then calculate and simplify the matrix product
\[A_nA_{n-1}A_{n-2}\cdots A_2.\]

(Tokyo University Entrance Exam 2007)
 
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Determine a Matrix From Its Eigenvalue

Problem 259

Let
\[A=\begin{bmatrix}
a & -1\\
1& 4
\end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is some real number.
Suppose that the matrix $A$ has an eigenvalue $3$.

(a) Determine the value of $a$.

(b) Does the matrix $A$ have eigenvalues other than $3$?

 
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Rotation Matrix in Space and its Determinant and Eigenvalues

Problem 218

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 orthogonal matrix.

(c) Find the eigenvalues of $A$.

 
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Maximize the Dimension of the Null Space of $A-aI$

Problem 200

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 dimension times five.

(The Ohio State University Linear Algebra Practice Problem)
 
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Find Values of $h$ so that the Given Vectors are Linearly Independent

Problem 194

Find the value(s) of $h$ for which the following set of vectors
\[\left \{ \mathbf{v}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
h \\
1 \\
-h
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
1 \\
2h \\
3h+1
\end{bmatrix}\right\}\] is linearly independent.

(Boston College, Linear Algebra Midterm Exam Sample Problem)
 
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Compute Determinant of a Matrix Using Linearly Independent Vectors

Problem 193

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
\[A\mathbf{x}=\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}, A\mathbf{y}=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, A\mathbf{z}=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}.\]

Then find the value of the determinant of the matrix $A$.

 
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Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam)

Problem 178

Let
\[\begin{bmatrix}
0 & 0 & 1 \\
1 &0 &0 \\
0 & 1 & 0
\end{bmatrix}.\]

(a) Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers?

(b) Calculate $A^{2009}$.

(Princeton University, Linear Algebra Exam)
 
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