Exercises 14 (Inverse matrix method and Cramer’s rule)

  1. Solve the first two pairs of equations using Cramer’s rule and use a matrix method for the third:

    1. \(\left\{\begin{aligned}3x-7y&=-2\\4x-3y&=7\end{aligned}\right.\)
    2. \(\left\{\begin{aligned}4x-3y&=18\\ x+2y&=-1\end{aligned}\right.\)
    3. \(\left\{\begin{aligned}11x-10y&=30\\21y-20x&=-40\end{aligned}\right.\)

  2. The equations of two straight lines are \(6x – 8.5y = 10\) and \(2x - 4y = 8\). By using a determinant method, establish the coordinates where the two lines cross.

  3. The following circuit yields a pair of simultaneous equations: \[\begin{align*} 5I_1 - 10I_2 &=0\\ 6(I_1+I_2)+5I_2 &=12 \end{align*}\] Determine the values of \(I_1\) and \(I_2\) using a matrix method.

  4. Solve the equations using a determinant method: \[\begin{align*} \frac{3}{x}-\frac{2}{y} &= 0.5\\ \frac{5}{x}-\frac{3}{y} &= 2.57 \end{align*}\]

  5. A vector system to determine the shortest distance between two moving bodies is analysed and produces the following equations: \[\begin{align*} 11s_1 - 10s_2 &= 30\\ 21s_2 - 20s_1 &= -40 \end{align*}\] Using Cramer’s rule solve for \(s_1\) and \(s_2\).

  6. The law connecting friction, \(F\), and load, \(L\), for an experiment to establish the friction force between two surfaces is of the form \(F = aL + b\), where both \(a\) and \(b\) are constants. When \(F = 6\), \(L = 7.5\) and when \(F = 2.7\), \(L = 2\), determine the values of \(a\) and \(b\) using a matrix method.

Solutions: 1. (i) \(x=55/19\), \(y=29/19\); (ii) \(x=3\), \(y=-2\); (iii) \(x=230/31\), \(y=160/31\); 2. \(x=-4\), \(y=-4\); 3. \(I_1=24/23\), \(I_2=12/23\); 4. \(x=0.27\), \(y=0.19\); 5. \(s_1=230/31\), \(s_2=160/31\); 6. \(a=0.6\), \(b=1.5\)