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Online quiz (due Wednesday 3rd May, 11:59pm):

Answer all five questions. Each question is worth 2 marks, the total is 10 marks.

Q1.

– POWERS

Let $A=\begin{pmatrix}1&1&1\\ 0&1&1\\ 0&0&1\end{pmatrix}\in\operatorname{M}_{{3}}(\mathbb{R})$ . Then $A^{5}$ is the matrix:
- (a)
  
  $\begin{pmatrix}5&5&5\\ 0&5&5\\ 0&0&5\end{pmatrix}$ ,
- (b)
  
  $\begin{pmatrix}1&5&25\\ 0&1&5\\ 0&0&1\end{pmatrix}$ ,
- (c)
  
  $\begin{pmatrix}1&5&15\\ 0&1&5\\ 0&0&1\end{pmatrix}$ ,
- (d)
  
  $\begin{pmatrix}1&5&10\\ 0&1&5\\ 0&0&1\end{pmatrix}$ ,
- (e)
  
  $\begin{pmatrix}5&10&15\\ 0&5&10\\ 0&0&5\end{pmatrix}$ .
Q2.

– SYMMETRIC

How many of the following five statements are true?
- –
  
  Let $A$ be a skew-symmetric matrix. Then $AA^{t}$ is symmetric.
- –
  
  Let $A$ be a skew-symmetric matrix. Then $A^{2}$ is symmetric.
- –
  
  Let $A$ be a symmetric matrix. Then $A^{2}$ is skew-symmetric.
- –
  
  Let $A$ be a symmetric matrix. Then $-A$ is symmetric.
- –
  
  Let $A$ be a skew-symmetric matrix. Then $-A$ is skew-symmetric.
- (a)
  
  0
- (b)
  
  1
- (c)
  
  2
- (d)
  
  3
- (e)
  
  4
Q3.

– MULTIPLICATION

Let $A\in\operatorname{M}_{{2}}(\mathbb{R})\quad\hbox{{,}}\quad B\in\operatorname{M% }_{{3\times 2}}(\mathbb{R})\quad\hbox{{,}}\quad C\in\operatorname{M}_{{2\times 3% }}(\mathbb{R})\quad\hbox{{and}}\quad D\in\operatorname{M}_{{3}}(\mathbb{R})$ . How many possible products of any two of these matrices are there? The chosen two matrices can be the same.
- (a)
  
  7,
- (b)
  
  8,
- (c)
  
  9,
- (d)
  
  10,
- (e)
  
  none.
Q4.

– TRANSPOSE

Let $A=(a_{ij})\in\operatorname{M}_{{4}}(\mathbb{R})$ with $a_{ij}=j-i$ for all $1\leq i,j\leq 4$ . Then $(A+A^{t})^{2}$ is the matrix:
- (a)
  
  It is the zero $4\times 4$ matrix.
- (b)
  
  $\begin{pmatrix}0&1&4&9\\ 1&0&1&4\\ 4&1&0&1\\ 9&4&1&0\end{pmatrix}$ .
- (c)
  
  It is not defined.
- (d)
  
  $\begin{pmatrix}1&2&4&6\\ 2&4&6&8\\ 4&6&8&10\\ 6&8&10&12\end{pmatrix}$ .
- (e)
  
  $\begin{pmatrix}0&1&2&3\\ -1&0&1&2\\ -2&-1&0&1\\ -3&-2&-1&0\end{pmatrix}$ .
Q5.

– REDUCED ROW ECHELON

How many of the following matrices are in reduced row echelon form?

$\begin{pmatrix}1&0&0&0\\ 0&0&0&0\\ 0&1&2&0\\ 0&0&0&1\end{pmatrix},\begin{pmatrix}1&0&0&0\\ 0&1&0&2\\ 0&0&0&1\\ 0&0&0&0\end{pmatrix},\begin{pmatrix}0&0&0&0\\ 1&0&2&0\\ 0&1&1&0\\ 0&0&0&1\end{pmatrix},\begin{pmatrix}1&0&0&0\\ 0&1&1&0\\ 0&0&0&2\\ 0&0&0&0\end{pmatrix},\begin{pmatrix}1&0&0&0&0\\ 0&1&1&0&0\\ 0&0&0&1&2\\ 0&0&0&0&1\end{pmatrix}$
- (a)
  
  0
- (b)
  
  1
- (c)
  
  2
- (d)
  
  3
- (e)
  
  4
Q6.

– CONSTRUCTING A MATRIX (III)

Which of the following commands creates the matrix

$\left[\begin{array}[]{ccccc}2&17&32&47&62\\ 5&20&35&50&65\\ 8&23&38&53&68\\ 11&26&41&56&71\\ 14&29&44&59&74\\ \end{array}\right]$
- (a)
  
  matrix((2:74),5,5,byrow=TRUE)
- (b)
  
  matrix(3*(1:25) - 1, 5,5,byrow=TRUE)
- (c)
  
  matrix(rep(2:74,3),5,5,byrow=TRUE)
- (d)
  
  matrix(rep(c(2,17,32,47,62),5),5,5,byrow=FALSE)
- (e)
  
  matrix(seq(2,74,by=3),5,5,byrow=FALSE)