Solve simultaneous equations using the inverse matrix

Solving simultaneous equations with an inverse matrix

Write the simultaneous equations as AX=C, where the coefficient rows and unknown column follow a fixed order. Find the inverse of A, multiply it by C on the left and read the solution column in that same order

1. Form the matrix equation Place the coefficients in rows, the unknowns in a column and the constants in a column \(\displaystyle AX=C,\quad A=\begin{bmatrix}a&b\\c&d\end{bmatrix},\quad X=\begin{bmatrix}x\\y\end{bmatrix}\)
2. Find the determinant Calculate ad-bc and check that it is not zero before using the inverse \(\displaystyle \det(A)=ad-bc\ne0\)
3. Multiply on the left Because A is on the left of X, multiply both sides by A⁻¹ on the left \(\displaystyle AX=C\quad\Rightarrow\quad A^{-1}AX=A^{-1}C\quad\Rightarrow\quad X=A^{-1}C\)
4. Read the solution order The first and second entries of the solution column belong to the first and second unknowns in the stated order \(\displaystyle X=\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}r\\s\end{bmatrix}\quad\Rightarrow\quad x=r,\ y=s\)

Worked example

Solve the simultaneous equations using an inverse matrix

1. \(\displaystyle A=\begin{bmatrix}3&2\\1&1\end{bmatrix},\quad X=\begin{bmatrix}x\\y\end{bmatrix},\quad C=\begin{bmatrix}16\\6\end{bmatrix},\quad AX=C\)Use the coefficients as rows and keep x then y as the unknown order
2. \(\displaystyle \det(A)=3\times1-2\times1=1,\quad A^{-1}=\begin{bmatrix}1&-2\\-1&3\end{bmatrix}\)The determinant is 1, so the inverse has integer entries
3. \(\displaystyle X=A^{-1}C=\begin{bmatrix}1&-2\\-1&3\end{bmatrix}\begin{bmatrix}16\\6\end{bmatrix}=\begin{bmatrix}16-12\\-16+18\end{bmatrix}=\begin{bmatrix}4\\2\end{bmatrix}\)Multiply the inverse by the constant column on the left and add the two products in each row
4. \(\displaystyle x=4,\quad y=2\)Read the first and second entries of the solution column as x and y.

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