Solve the system of given equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. begin{cases}3a-b-4c=32a-b+2c=-8a+2b-3c=9end{cases}

Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.
${\displaystyle g\left(x\right)=2\sqrt{3}-x,(-\mathrm{\infty },3]}$

Use cramer's rule to determine the values of $I_1,I_2,I_3$ and $I_4$
$\left[\begin{array}{cccc}13.7& -4.7& -2.2& 0\\ -4.7& 15.4& 0& -8.2\\ -2.2& 0& 25.4& -22\\ 0& -8.2& -22& 31.3\end{array}\right]\left[\begin{array}{c}{I}_1\\ I_2\\ I_3\\ I_4\end{array}\right]=\left[\begin{array}{c}6\\ -6\\ 5\\ -9\end{array}\right]$

Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
$\{\begin{array}{l}{x}_1+x_2-3x_3=-1\\ -x_1+2x_2=1\\ x_1-x_2+x_3=2\end{array}$

Mark the statement True or False, and justify your answer. Two matrices are row equivalent if they have the same number of rows.

Let $A=\left[\begin{array}{ccc}1& 2& -1\\ 1& 0& 3\\ 1& 1& -1\end{array}\right]$
i) Find the determinant of A
ii)Verify that the matrix
$\left[\begin{array}{ccc}-\frac{3}{4}& \frac{1}{4}& \frac{6}{4}\\ 1& 0& -1\\ \frac{1}{4}& \frac{1}{4}& -\frac{1}{2}\end{array}\right]$ is the inverse matrix of A