Linear-algebra-quiz-eye-catch
Linear-algebra-quiz-eye-catch
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- Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even
Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.
Then prove the following statements.
(a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.
(b) The rank of $A$ is even.
Proof.
(a) Each […]
- The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization
Consider the $2\times 2$ real matrix
\[A=\begin{bmatrix}
1 & 1\\
1& 3
\end{bmatrix}.\]
(a) Prove that the matrix $A$ is positive definite.
(b) Since $A$ is positive definite by part (a), the formula
\[\langle \mathbf{x}, […]
- Quiz 13 (Part 1) Diagonalize a Matrix
Let
\[A=\begin{bmatrix}
2 & -1 & -1 \\
-1 &2 &-1 \\
-1 & -1 & 2
\end{bmatrix}.\]
Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.
That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
- Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57
Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group.
Then determine the number of elements in $G$ of order $3$.
Proof.
Observe the prime factorization $57=3\cdot 19$.
Let $n_{19}$ be the number of Sylow $19$-subgroups of $G$.
By […]
- Find a Basis for a Subspace of the Vector Space of $2\times 2$ Matrices
Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where
\begin{align*}
A_1=\begin{bmatrix}
1 & 2 \\
-1 & 3
\end{bmatrix}, \quad
A_2=\begin{bmatrix}
0 & -1 \\
1 & 4
[…]
- Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$
Let $A$ be an $n\times n$ nonsingular matrix with integer entries.
Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.
Hint.
If $B$ is a square matrix whose entries are integers, then the […]
- Describe the Range of the Matrix Using the Definition of the Range
Using the definition of the range of a matrix, describe the range of the matrix
\[A=\begin{bmatrix}
2 & 4 & 1 & -5 \\
1 &2 & 1 & -2 \\
1 & 2 & 0 & -3
\end{bmatrix}.\]
Solution.
By definition, the range $\calR(A)$ of the matrix $A$ is given […]
- Probabilities of An Infinite Sequence of Die Rolling
Consider an infinite series of events of rolling a fair six-sided die. Assume that each event is independent of each other. For each of the below, determine its probability.
(1) At least one die lands on the face 5 in the first $n$ rolls.
(2) Exactly $k$ dice land on the face 5 […]