To find the determinant of any 3x3 matrix \(\displaystyle a=\begin{bmatrix} a&b &c \\ d& e& f\\ g& h& i \end{bmatrix}\), we use the formula \(\displaystyle determinant=a(ei-hf)-b(di-gf)+c(dh-ge)\). Using the vector from the problem statement, we get \(\displaystyle 1(0-6)-3(35+9)-1(10-0)=-6-132-10=-148\)

To find the difference between two 2x2 matrices \(\displaystyle a=\begin{bmatrix} a& b\\ c&d \end{bmatrix}\) and \(\displaystyle b=\begin{bmatrix} e& f\\ g& h \end{bmatrix}\) you use the formula \(\displaystyle a-b=\begin{bmatrix} a-e&b-f \\ c-g& d-h \end{bmatrix}\). Using the two matrices from the problem statement, we then get \(\displaystyle \begin{bmatrix} 2-0& 1-7\\ 3-1& 6-5 \end{bmatrix}=\begin{bmatrix} 2&-6 \\ 2& 1 \end{bmatrix}\)

To find the difference between two 2x2 matrices \(\displaystyle a=\begin{bmatrix} a& b\\ c&d \end{bmatrix}\) and \(\displaystyle b=\begin{bmatrix} e& f\\ g& h \end{bmatrix}\) you use the formula \(\displaystyle a-b=\begin{bmatrix} a-e&b-f \\ c-g& d-h \end{bmatrix}\). Using the two matrices from the problem statement, we then get \(\displaystyle \begin{bmatrix} 4-1& x-5x\\ z-3z& 5-7 \end{bmatrix}=\begin{bmatrix} 3&-4x \\ -2z& -2\end{bmatrix}\)

To find the determinant of a 2x2 matrix \(\displaystyle a=\begin{bmatrix} a&b \\ c&d \end{bmatrix}\) ,we use the formula 

\(\displaystyle determinant=(ad)-(cb)\)

Using hte matrix from the problem statement, we get

\(\displaystyle determinant=(-4*5)-(1*2)=-22\) 

To find the determinant of the 3x3 matrix \(\displaystyle \begin{bmatrix} i& j& k\\ a&b &c \\ x&y &z \end{bmatrix}\), we use the formula:

\(\displaystyle determinant=i(bz-yc)-j(az-xc)+k(ay-xb)\). Using the matrix from the problem statement, we get:

\(\displaystyle i(-20-0)-j(15-2)+k(0+8)=\left \langle -20,-13,8\right \rangle\)

To find the determinant of the 3x3 matrix \(\displaystyle \begin{bmatrix} i& j& k\\ a&b &c \\ x&y &z \end{bmatrix}\), we use the formula:

\(\displaystyle determinant=i(bz-yc)-j(az-xc)+k(ay-xb)\). Using the matrix from the problem statement, we get:

\(\displaystyle i(7-6)-j(14-6)+k(12-6)=\left \langle 1,-8,6\right \rangle\)

To do matrix summation, you apply the following formula, where a and b are 4x4 matrices: \(\displaystyle \begin{bmatrix} a&b &c &d \\ e&f &g &h \\ i&j &k &l \\ m&n & o&p \end{bmatrix}+\begin{bmatrix} q& r& s&t \\ u&v &w &x \\ y&z &a_1 &b_1 \\ c_1&d_1 &e_1 &f_1 \end{bmatrix}=\begin{bmatrix} a+q&b+r &c+s &d+t \\ e+u&f+v &g+w &h+x \\ i+y&j+z &k+a_{1} &l+b_{1} \\ m+c_{1}&n+d_{1} &o+e_{1} &p+f_{1} \end{bmatrix}\)

Using this formula, we get \(\displaystyle \begin{bmatrix} 8&4 &8 &1 \\ 8&5 &2 &13 \\ 8&15 &15 &3 \\ 6&9 &9 &6 \end{bmatrix}\)

To find the the determinant of a 3x3 matrix \(\displaystyle a=\begin{bmatrix} a& b&c \\ d&e &f \\ h&i &j \end{bmatrix}\), you apply the following formula

\(\displaystyle determinant=a(ej-if)-b(dj-hf)+c(di-he)\)

Applying to the matrix from the problem statement, we get

\(\displaystyle 1(42-0)-4(18-40)+7(0-35)=-115\)

To find the the determinant of a 3x3 matrix \(\displaystyle a=\begin{bmatrix} a& b&c \\ d&e &f \\ h&i &j \end{bmatrix}\), you apply the following formula

\(\displaystyle determinant=a(ej-if)-b(dj-hf)+c(di-he)\)

Applying to the matrix from the problem statement, we get

\(\displaystyle 2(7-12)-0(21-30)+4(6-5)=-6\)

To find the determinant of a 2x2 matrix \(\displaystyle a=\begin{bmatrix} x&y \\ z&q \end{bmatrix}\), we use the formula:

\(\displaystyle detemrinant=xq-zy\)

Using the vector from the problem statement, we get:

\(\displaystyle (3*2)-(7*x)=6-7x\)