Questions

1. Question 1

   Find the inverses of the following matrices by using row operations:

   (i)

   $\begin{bmatrix}2&6&-3\\0&-2&0\\-2&5&6\end{bmatrix}$

   (ii)

   $\begin{bmatrix}1&2&-1\\0&-2&8\\1&0&2\end{bmatrix}$

   (iii)

   $\begin{bmatrix}1&6&2\\2&13&0\\0&-1&1\end{bmatrix}$

   SolutionTheory
2. Question 2

   Find the rank of the following matrices:

   (i)

   $\begin{bmatrix}1&-1&3&1\\-2&-6&1&-1\\3&1&4&-2\end{bmatrix}$

   (ii)

   $\begin{bmatrix}1&-2&3\\2&-4&6\\-1&0&2\\0&1&-1\end{bmatrix}$

   (iii)

   $\begin{bmatrix}3&-1&3&0&1\\1&2&-1&-3&-2\\2&3&4&2&1\\2&5&-2&-3&3\end{bmatrix}$

   SolutionTheory
3. Question 3

   Solve the following systems of linear equations by Cramer’s rule:

   (i)

   $2x+y-z=1,\; x-y+2z=3,\; 3x+2y+z=4$

   (ii)

   $x_1+2x_2-3x_3=0,\; 4x_1-x_2+x_3=5,\; -2x_1+3x_2+2x_3=3$

   (iii)

   $2x_1-x_2+x_3=1,\; x_1+2x_2+2x_3=2,\; x_1-2x_2-x_3=1$

   SolutionTheory
4. Question 4

   Solve the following systems of linear equations by matrix inversion method:

   (i)

   $x-2y+z=-1,\; 3x+y-2z=4,\; y-z=1$

   (ii)

   $2x_1+x_2+3x_3=3,\; x_1+3x_2-2x_3=0,\; -3x_1-x_2+2x_3=4$

   (iii)

   $x+y=2,\; 2x-z=1,\; 2y-3z=-1$

   SolutionTheory
5. Question 5

   Solve the following systems by reducing their augmented matrices to the echelon form and the reduced echelon forms:

   (i)

   $x_1+2x_2-2x_3=-1,\; 2x_1+3x_2+x_3=1,\; 5x_1+4x_2-3x_3=1$

   (ii)

   $x+2y+z=2,\; 2x+y+2z=3,\; 2x+3y-z=7$

   (iii)

   $x_1+4x_2+x_3=2,\; 2x_1+x_2-2x_3=9,\; 3x_1+x_2-x_3=12$

   SolutionTheory
6. Question 6

   Solve the following systems of homogeneous linear equations by using Gaussian elimination method:

   (i)

   $x+4y-2z=0,\; 2x+y+5z=0,\; 5x+2y+8z=0$

   (ii)

   $x_1+4x_2+2x_3=0,\; 2x_1+x_2-3x_3=0,\; 3x_1+2x_2-4x_3=0$

   (iii)

   $x_1+2x_2-x_3=0,\; x_1-x_2+5x_3=0,\; 2x_1+x_2+4x_3=0$

   SolutionTheory
7. Question 7

   A triangle has vertices at $A(4,1)$, $B(-2,5)$ and $C(0,-3)$. Find the vertices of the reflected triangle over the $y$-axis using a transformation matrix.

   SolutionTheory
8. Question 8

   The point $A$ is mapped to $(30,20,-5)$ by the scaling matrix $P=\begin{bmatrix}-5&0&0\\0&-5&0\\0&0&-5\end{bmatrix}$. Find the coordinates of $A$.

   SolutionTheory
9. Question 9

   Find the equation of the image of the curve with equation $y=x^2$ under the transformation with associated matrix $\begin{bmatrix}1&2\\3&4\end{bmatrix}$.

   SolutionTheory
10. Question 10

    Use the matrix $A=\begin{bmatrix}1&0&1\\2&-1&1\\0&1&2\end{bmatrix}$ to encode the message: KEEP IT UP, where letters A to Z are corresponding to the numbers 1 to 26.

    SolutionTheory
11. Question 11

    Decode the message $\begin{bmatrix}11&25&22\\20&10&14\\43&41&41\end{bmatrix}$ that was encoded using matrix $A=\begin{bmatrix}1&1&-1\\1&0&1\\2&1&1\end{bmatrix}$, where the numbers 1 to 26 are corresponding to the letters A to Z, and 27 is representing space or "-".

    SolutionTheory