# The electrical currents flowing in the antennas, during an MR imaging

The electrical currents flowing in the antennas, during an MR imaging scan are given by:
$3{i}_{3}+3{i}_{1}-5{i}_{2}=7.5$
${i}_{2}-7{i}_{3}+2{i}_{1}=-17.5$
$-10{i}_{1}+4{i}_{2}+5{i}_{3}=16$
1. Write the above system of equations in the form $Ax=b$, where A is a 3 by 3 matrix and x and b are 3 by 1 vectors.
2. Find the determinant of A.
3. Determine and ${i}_{3}$ by using Gaussian elimination.
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Step 1
$\left[\begin{array}{ccc}3& -5& 3\\ 2& 1& -7\\ -10& 4& 5\end{array}\right]\left[\begin{array}{c}{i}_{1}\\ {i}_{2}\\ {i}_{3}\end{array}\right]=\left[\begin{array}{c}7.5\\ -17.5\\ 16\end{array}\right]$
above solution is form if
$Ax=b$
where
$A=\left[\begin{array}{ccc}3& -5& 3\\ 2& 1& -7\\ -10& 4& 5\end{array}\right]$

Step 2
Determinant of A
$D\left[\begin{array}{ccc}3& -5& 3\\ 2& 1& -7\\ -10& 4& 5\end{array}\right]$
$D\left(A\right)=3\left(5-\left(-28\right)\right)-\left(-5\right)\left(10-70\right)$
$+3\left(8-\left(10\right)\right)$
$=3\left(33\right)+5\left(-60\right)+3\left(18\right)$
$=99-300+54$
$=153-300$
$D\left(A\right)=-147$
Step 3
Now we have to calculate and find using gaussian elimination
Solve it using gaussian elimination
$\left[\begin{array}{cccc}3& -5& 3& 7.5\\ 2& 1& -7& -17.5\\ -10& 4& 5& 16\end{array}\right]$
$\frac{{R}_{1}}{3}\to {R}_{1}$
$\left[\begin{array}{cccc}1& -\frac{5}{3}& 1& 2.5\\ 2& 1& -7& -17.5\\ -10& 4& 5& 16\end{array}\right]$
${R}_{2}-2{R}_{1}\to {R}_{2}$
${R}_{3}+10{R}_{1}\to {R}_{3}$
$\left[\begin{array}{cccc}1& -\frac{5}{3}& 1& 2.5\\ 0& \frac{13}{3}& -9& -22.5\\ 0& -\frac{38}{3}& 15& 41\end{array}\right]$
$\frac{{R}_{1}}{\frac{13}{3}}\to {R}_{2}$
${R}_{1}+\frac{5}{3}{R}_{2}\to {R}_{1}$
${R}_{3}+\frac{38}{3}{R}_{2}\to {R}_{3}$