Writing equations in matrix form
A pair of simultaneous equations can be written as A × X = C. The coefficient matrix stores one signed coefficient row for each equation, while X and C are separate columns
\[A\times X=C\]
- Declare the variable order List the unknowns in a fixed order before placing their coefficients \(\displaystyle X=\begin{bmatrix}x\\y\end{bmatrix}\)
- Use one row per equation The coefficients in the first equation form row 1 and those in the second form row 2 \(\displaystyle 5x-2y=11\Rightarrow\begin{bmatrix}5&-2\end{bmatrix}\)
- Keep coefficient signs Move a term across the equals sign before reading its coefficient, changing its sign when it moves \(\displaystyle 2x=7-y\Rightarrow2x+y=7\Rightarrow\begin{bmatrix}2&1\end{bmatrix}\)
- Separate constants Put the unknowns in X and the constants in the column C \(\displaystyle \begin{bmatrix}5&-2\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}11\\19\end{bmatrix}\)
Worked example
Write the following simultaneous equations as A × X = C, using the variable order x, y
\[5x-2y=11,\quad3x+4y=19\]
- \(\displaystyle A=\begin{bmatrix}5&-2\\3&4\end{bmatrix}\)Put the signed coefficients from each equation into its own row
- \(\displaystyle X=\begin{bmatrix}x\\y\end{bmatrix},\quad C=\begin{bmatrix}11\\19\end{bmatrix}\)Use the declared unknown order for X and put the constants in a separate column
- \(\displaystyle \begin{bmatrix}5&-2\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}11\\19\end{bmatrix}\)Multiplying each coefficient row by the unknown column reproduces the two original equations
Common mistakes
Putting equations in columns
✕\(\displaystyle \begin{bmatrix}5&3\\-2&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}11\\19\end{bmatrix}\)✓\(\displaystyle \begin{bmatrix}5&-2\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}11\\19\end{bmatrix}\)
Each equation supplies one row, so placing coefficients in columns mixes the two equationsDropping a coefficient sign
✕\(\displaystyle 5x-2y=11\Rightarrow\begin{bmatrix}5&2\end{bmatrix}\)✓\(\displaystyle 5x-2y=11\Rightarrow\begin{bmatrix}5&-2\end{bmatrix}\)
The negative sign belongs to the coefficient of y and must stay in the matrixMixing constants and unknowns
✕\(\displaystyle \begin{bmatrix}5&-2\\3&4\end{bmatrix}\begin{bmatrix}11\\19\end{bmatrix}=\begin{bmatrix}x\\y\end{bmatrix}\)✓\(\displaystyle \begin{bmatrix}5&-2\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}11\\19\end{bmatrix}\)
The unknown column is multiplied by the coefficient matrix and the constants form the result columnDifficulty levels
introWrite two equations with positive coefficients as a coefficient matrix and constant column
standardRead a matrix equation back into equations while preserving negative coefficients
challengeHandle a changed variable order, rearranged terms and matrix-form recognition
examSelect the matrix equation whose rows and signs reproduce the given equations
Questions about this topic
Why does each equation become a row?+
A row of the coefficient matrix multiplied by the unknown column gives one linear expression, so each row reproduces one equation
What happens when a variable term is on the right side?+
Move it to the left before recording its coefficient, and change its sign as it crosses the equals sign
Why must the variable order be declared?+
The positions in the unknown column match the coefficient columns, so changing the order without changing the coefficients represents a different pair of equations
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices
In this chapter
State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals