Solving simultaneous equations with matricesAQA A-Level Further Maths: Revision notes

Three linear equations in $x$, $y$, $z$ can be written $\mathbf{A}\mathbf{x}=\mathbf{b}$, where the rows of $\mathbf{A}$ are the coefficients, $\mathbf{x}=\begin{pmatrix}x\\ y\\ z\end{pmatrix}$ and $\mathbf{b}$ holds the right-hand sides in the same order. If $\det\mathbf{A}\neq0$ then $\mathbf{A}^{-1}$ exists, and multiplying on the left gives $\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}.$ Example: $x+y+z=6$, $2x-y+z=6$, $x+3y-z=2$ has $\mathbf{A}^{-1}=\frac18\begin{pmatrix}-2&4&2\\3&-2&1\\7&-2&-3\end{pmatrix}$, so $\mathbf{x}=\frac18\begin{pmatrix}-2&4&2\\3&-2&1\\7&-2&-3\end{pmatrix}\begin{pmatrix}6\\6\\2\end{pmatrix}=\begin{pmatrix}2\\1\\3\end{pmatrix}$. Always check the solution in one of the original equations.

If $\det\mathbf{A}=0$ the matrix is singular, there is no inverse, and the system does not have exactly one solution. There are two possibilities:

• Consistent: infinitely many solutions.
• Inconsistent: no solutions. To decide which, use elimination. If combining equations produces a true statement for every value (such as $0=0$), the system is consistent. If it produces a false statement (such as $5=7$), it is inconsistent. Example: for $x+2y+z=4$, $2x+y-z=1$, $3x+3y=k$, adding the first two gives $3x+3y=5$. When $k=5$ the third equation is redundant, so there are infinitely many solutions; when $k\neq5$ there is a contradiction and no solution.

Each equation $px+qy+rz=s$ is a plane. A solution is a point on all three planes. Unique solution ($\det\neq0$): the three planes meet at a single point. Infinitely many solutions ($\det=0$, consistent):

• three planes meeting in a line (a sheaf, like pages in a book)
• two planes coincide and the third crosses them in a line
• all three planes coincide (solutions fill the plane). No solution ($\det=0$, inconsistent):
• a triangular prism: the planes meet in pairs in three parallel lines, with no common point
• two parallel planes with a third crossing them
• three parallel planes (distinct)
• two coincident planes with a third parallel to them.

Consider $x+y+z=1$, $x+2y+3z=3$, $2x+3y+az=b$. The determinant is $a-4$.

• $a\neq4$: unique solution, three planes meet at a point. For $a=5$, $b=7$ the solution is $(2,-4,3)$.
• $a=4$: adding the first two equations gives $2x+3y+4z=4$. If $b=4$, the third equation is redundant, and with $z=t$ the solution is $(-1+t,\,2-2t,\,t)$, a line (sheaf).
• $a=4$, $b\neq4$: contradiction, so no solution; no two planes are parallel, so the planes form a triangular prism.

• State clearly the matrix equation you are solving, and which side the inverse multiplies.
• Use a calculator for inverses only when the question allows it; otherwise use elimination.
• For explain questions give both the algebra (determinant zero or non-zero) and the geometry.
• Parametrise infinite solutions with $t$, and give all three coordinates in terms of $t$.
• When asked about the geometry, name the arrangement: single point, line (sheaf), triangular prism, parallel planes.