Question 9.19: Use the inverse matrix to solve the following simultaneous e......

Step 1: Write all the equations in the same order: variables (in order) = RHS. Write down the matrices A and B. Thematrix A is the matrix of coefficients of the three equations, all arranged in the same format; B is the column matrix consisting of the constants from the RHS of the equations.

A = \left(\begin{matrix} 1 & 0 & -2 \\ 2 & 2 & 3 \\ 1 & 3 & 2 \end{matrix}  \right)         B = \left(\begin{matrix} 4 \\ 15\\ 12 \end{matrix}  \right)

Step 2: Since X = A^{−1} B, determine the inverse of A. However, if you look carefully at A, you will see that this matrix is identical to the matrix D in Worked Examples 9.17 and 9.18 in which the inverse of D was determined; therefore

A^{−1} =\frac{1}{-13} \left(\begin{matrix} -5 & -6 & 4 \\ -1 & 4 & -7 \\ 4 & -3 & 2 \end{matrix}  \right)

So proceed straight away to the next step.

Step 3: Premultiply the matrix B by the inverse of A:

X = A^{−1} B

\left(\begin{matrix} x \\ y\\ z \end{matrix} \right) =\frac{1}{-13} \left(\begin{matrix} -5 & -6 & 4 \\ -1 & 4 & -7 \\ 4 & -3 & 2 \end{matrix} \right) \left(\begin{matrix} 4 \\ 15\\ 12 \end{matrix}\right) = -\frac{1}{13} \left(\begin{matrix} -62 \\ -28\\ -5 \end{matrix}\right) = \left(\begin{matrix} 4.77 \\ 2.15\\ 0.38 \end{matrix}\right)

\left(\begin{matrix} x \\ y\\ z \end{matrix} \right) = \left(\begin{matrix} 4.77 \\ 2.15\\ 0.38 \end{matrix}\right)

The solutions may simply be read off (correct to two decimal places), x = 4.77,
y = 2.15, z = 0.38.