Question

# Use cramer's rule to determine the values of I_1, I_2, I_3 and I_4 begin{bmatrix}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{bmatrix}begin{bmatrix}I_1 I_2 I_3 I_4 end{bmatrix}=begin{bmatrix}6 -6 5 -9 end{bmatrix}

Matrices
Use cramer's rule to determine the values of $$I_1, I_2, I_3$$ and $$I_4$$
$$\begin{bmatrix}13.7 & -4.7 & -2.2 &0 \\ -4.7 & 15.4 & 0 &-8.2 \\-2.2 & 0 & 25.4 &-22 \\ 0 & -8.2 & -22 &31.3 \end{bmatrix}\begin{bmatrix}I_1 \\ I_2 \\ I_3 \\ I_4 \end{bmatrix}=\begin{bmatrix}6 \\ -6 \\ 5 \\-9 \end{bmatrix}$$

2021-03-12
Step 1
Given matrices:
$$A=\begin{bmatrix}13.7 & -4.7 & -2.2 &0 \\ -4.7 & 15.4 & 0 &-8.2 \\-2.2 & 0 & 25.4 &-22 \\ 0 & -8.2 & -22 &31.3 \end{bmatrix},B=\begin{bmatrix}I_1 \\ I_2 \\ I_3 \\ I_4 \end{bmatrix},C=\begin{bmatrix}6 \\ -6 \\ 5 \\-9 \end{bmatrix}$$
Multiplication of matrices A and B is possible only when number of columns of matrix A is equal to number of rows of matrix B.
Here order of matrix A is $$4 \times 4$$ and order of matrix B is $$4 \times 1$$ and hence the resultant matrix C has order $$4 \times 1$$
Step 2
Multiplication of matrices:
$$\begin{bmatrix}13.7 & -4.7 & -2.2 &0 \\ -4.7 & 15.4 & 0 &-8.2 \\-2.2 & 0 & 25.4 &-22 \\ 0 & -8.2 & -22 &31.3 \end{bmatrix}\begin{bmatrix}I_1 \\ I_2 \\ I_3 \\ I_4 \end{bmatrix}=\begin{bmatrix}6 \\ -6 \\ 5 \\-9 \end{bmatrix}$$
$$\begin{bmatrix}(13.7)I_1-4.7I_2-2.2I_3+(0)I_4 \\ -4.7I_1+15.4I_2 + (0)I_3-8.2I_4 \\-2.2I_1+(0)I_2+25.4I_3-22I_4 \\ (0)I_1-8.2I_2 -22I_3+31.3I_4 \end{bmatrix}=\begin{bmatrix}6 \\ -6 \\ 5 \\-9 \end{bmatrix}$$
From the above equation of matrix, four simultaneous equations obtained are:
$$13.7I_1-4.7I_2-2.2I_3=6$$
$$-4.7I_1+15.4I_2-8.2I_4=-6$$
$$-2.2I_1+25.4I_3-22I_4=5$$
$$-8.2I_2 -22I_3+31.3I_4=-9$$
Solve the above simultaneous equations to get values of unknown variables
$$I_1=0.021$$
$$I_2=-0.908$$
$$I_3=-0.655$$
$$I_4=-0.986$$