Writing a system of linear equations as a matrix equation, solving by the inverse matrix, and the geometric interpretation of consistent, inconsistent and dependent systems in two and three unknowns.

What this dot point is asking

OCR wants you to express a system of simultaneous linear equations as a single matrix equation $\mathbf{M}\mathbf{x} = \mathbf{b}$, solve it by multiplying through by $\mathbf{M}^{-1}$ when $\mathbf{M}$ is non-singular, and interpret geometrically what happens when $\det \mathbf{M} = 0$: in three unknowns the equations are planes, and you must distinguish a unique intersection point, a common line (a sheaf), and an inconsistent arrangement with no common point.

Writing a system as a matrix equation

Any set of linear equations in the same unknowns can be packed into a matrix equation, separating the coefficients from the unknowns and the constants. This is the form that lets a single inverse solve the whole system at once.

Solving by the inverse matrix

When $\mathbf{M}$ is non-singular, multiply both sides on the left by $\mathbf{M}^{-1}$. Because $\mathbf{M}^{-1}\mathbf{M} = \mathbf{I}$, the unknowns are isolated.

The geometry when the determinant is zero

In three unknowns each equation is a plane. The value of the determinant decides how the three planes sit relative to one another.

• Non-singular ($\det \mathbf{M} \ne 0$). The planes meet at exactly one point; a unique solution.
• Singular and consistent. The planes share a common line. This is a sheaf (the planes pivot about a common line) and there are infinitely many solutions. A special sub-case has two or three planes coincide.
• Singular and inconsistent. The planes have no common point. Either two planes are parallel, or the three form a triangular prism (each pair meets in a line, but the three lines are distinct and parallel). There is no solution.

To classify a singular system, eliminate one unknown between pairs of equations and compare: if you reach a true identity or a single relation, the system is a sheaf; if you reach a contradiction such as $0 = 5$, it is a prism with no solution.

This links the algebra of matrices to the geometry of planes, which is developed further in the vectors strand.

Try this

Q1. Solve $\begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 11 \end{pmatrix}$ by the inverse method. [3 marks]

• Cue. $\det = -1$, inverse $\begin{pmatrix} -5 & 2 \\ 3 & -1 \end{pmatrix}$, so $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$.

Q2. A $3 \times 3$ coefficient matrix has determinant $0$ and elimination gives $0 = 0$. What is the geometric configuration of the three planes? [2 marks]

• Cue. Consistent and singular, so the planes form a sheaf meeting in a common line (infinitely many solutions).