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Online quiz (due Wednesday 10th May, 11:59pm):

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

Q1.

– ELEMENTARY PRODUCTS

In $\operatorname{M}_{{3}}(\mathbb{R})$ , calculate the product

$E_{2}(2)E_{13}(4)E_{13}E_{12}(-1)E_{1}(3).$
- (a)
  
  $\begin{pmatrix}12&-1&1\\ 0&2&0\\ 3&-1&0\end{pmatrix}$ ,
- (b)
  
  $\begin{pmatrix}0&-6&3\\ 0&2&0\\ 1&0&4\end{pmatrix}$ ,
- (c)
  
  $\begin{pmatrix}-12&-4&-1\\ 0&2&0\\ 3&-1&0\end{pmatrix}$ ,
- (d)
  
  $\begin{pmatrix}12&-4&1\\ 0&2&0\\ 3&-1&0\end{pmatrix}$ ,
- (e)
  
  $\begin{pmatrix}12&-4&-1\\ 0&2&0\\ 3&-1&0\end{pmatrix}$ .
Q2.

– INVERSE

Let $A=\begin{pmatrix}1&1&1&1\\ 1&0&1&2\\ -2&-1&1&0\\ 1&0&0&2\end{pmatrix}$ . Then ${A}^{-1}$ is the matrix:
- (a)
  
  $\begin{pmatrix}-\frac{2}{3}&\frac{4}{3}&-\frac{2}{3}&1\\ \frac{4}{3}&-\frac{5}{3}&\frac{1}{3}&1\\ 0&1&0&-1\\ \frac{1}{3}&\frac{2}{3}&\frac{1}{3}&1\end{pmatrix}$ ,
- (b)
  
  $\begin{pmatrix}-\frac{2}{3}&\frac{4}{3}&-\frac{2}{3}&-1\\ \frac{4}{3}&-\frac{5}{3}&\frac{1}{3}&1\\ 0&1&0&-1\\ \frac{1}{3}&-\frac{2}{3}&\frac{1}{3}&1\end{pmatrix}$ ,
- (c)
  
  $\begin{pmatrix}-\frac{2}{3}&\frac{4}{3}&-\frac{2}{3}&-1\\ \frac{4}{3}&-\frac{5}{3}&\frac{1}{3}&1\\ 0&\frac{1}{3}&0&-\frac{1}{3}\\ \frac{1}{3}&-\frac{2}{3}&\frac{1}{3}&\frac{1}{3}\end{pmatrix}$ ,
- (d)
  
  $\begin{pmatrix}-\frac{2}{3}&\frac{4}{3}&\frac{2}{3}&-1\\ \frac{4}{3}&-\frac{4}{3}&\frac{1}{3}&1\\ 0&1&0&-1\\ \frac{1}{3}&\frac{2}{3}&\frac{1}{3}&1\end{pmatrix}$ ,
- (e)
  
  $A$ is not invertible.
Q3.

– DETERMINANTS 1

The determinant of the matrix $A=\frac{1}{5}\begin{pmatrix}-1&3&-1\\ -3&0&2\\ 1&-2&-1\end{pmatrix}$ is equal to:
- (a)
  
  $\frac{13}{125}$ ,
- (b)
  
  $\frac{13}{5}$ ,
- (c)
  
  $-\frac{13}{125}$ ,
- (d)
  
  $-\frac{13}{5}$ ,
- (e)
  
  $13$ .
Q4.

– DETERMINANTS 2

How many different (real) values of $\lambda$ are there for which $\det(A-\lambda\cdot I_{4})=0$ , where

$A=\begin{pmatrix}1&1&0&1\\ -1&3&0&-2\\ -2&-1&1&0\\ 0&0&0&2\end{pmatrix}.$
- (a)
  
  $0$ ,
- (b)
  
  $1$ ,
- (c)
  
  $2$ ,
- (d)
  
  $3$ ,
- (e)
  
  $4$ .
Q5.

– QUADRATIC EQUATION

How many matrices $A\in M_{2}(\mathbb{R})$ satisfy the equation $A^{2}+A+I_{2}=0$ ?
- (a)
  
  $0$ ,
- (b)
  
  $1$ ,
- (c)
  
  $2$ ,
- (d)
  
  $4$ ,
- (e)
  
  Infinitely many.
Q6.

– MATRIX MULTIPLICATION

For $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$ and $\mathbf{E}$ as defined by A,B,C,D and E at the start of this weeks lab sheet, find

$\mathbf{CAB}^{T}\mathbf{D}+\mathbf{ECD}$
The result is
- (a)
  
  $\left[\begin{array}[]{cc}2456&1237\\ 5412&3112\end{array}\right]$
- (b)
  
  $\left[\begin{array}[]{cc}2456&5412\\ 1237&3112\end{array}\right]$
- (c)
  
  $\left[\begin{array}[]{cc}3525&2157\\ 5456&3536\end{array}\right]$
- (d)
  
  $\left[\begin{array}[]{cc}3525&5456\\ 2157&3112\end{array}\right]$
- (e)
  
  $\left[\begin{array}[]{cc}5321&2654\\ 3345&3411\end{array}\right]$