Systems of linear equations and larger matrices in QCE Specialist Mathematics Unit 3

What this dot point is asking

QCAA wants you to handle matrices larger than $2\times 2$, especially to represent a system of three linear equations in three unknowns as $A\mathbf{x} = \mathbf{b}$, solve it using the inverse $\mathbf{x} = A^{-1}\mathbf{b}$, and interpret what happens when the determinant is zero. The geometric reading, three planes meeting in a point, in a line or not at all, ties the algebra to the vectors topic.

The answer

Matrix form of a system

A system such as

is written compactly as $A\mathbf{x} = \mathbf{b}$, where $A$ is the $3\times 3$ matrix of coefficients, $\mathbf{x} = (x, y, z)^{\mathsf T}$ and $\mathbf{b} = (d_1, d_2, d_3)^{\mathsf T}$.

Determinant of a three by three matrix

Expanding along the first row,

The alternating signs ($+, -, +$) are essential. The determinant decides invertibility: $A$ has an inverse if and only if $\det A \neq 0$.

Inverse and solving

When $\det A \neq 0$ the system has a unique solution

In practice the inverse of a $3\times 3$ matrix is found from the adjugate divided by the determinant, or computed with technology, which QCAA permits in the relevant assessment. The point is to recognise the structure and read off the unique solution.

Geometric interpretation

Each equation $ax + by + cz = d$ is a plane. Solving the system locates the common points of three planes.

• Unique solution ($\det A \neq 0$): the three planes meet at a single point.
• No solution ($\det A = 0$, system inconsistent): the planes have no common point, for example two are parallel, or they form a triangular prism arrangement.
• Infinitely many solutions ($\det A = 0$, system consistent): the planes share a common line, or all coincide, giving a one-parameter family of solutions.

Why the determinant matters

A zero determinant signals that the rows (or the normal vectors of the planes) are linearly dependent, so the inverse method fails. You then test consistency directly by elimination to decide between no solution and infinitely many.

Solving by elimination (row reduction)

When the inverse is not used, Gaussian elimination is the systematic alternative. Use one equation to eliminate a variable from the others, then repeat, reducing the system to an upper-triangular form that can be back-substituted. If elimination produces a contradictory row such as $0 = 5$, the system is inconsistent and has no solution. If it produces an all-zero row such as $0 = 0$, one equation was dependent on the others, leaving a free variable and infinitely many solutions. This procedure works whether or not the determinant is zero, and it is the tool that distinguishes the two degenerate cases.

Parametrising an infinite solution set

When there are infinitely many solutions, express the answer with a parameter. Solve for two variables in terms of the third, set the free variable equal to $t$, and write the solution as $\mathbf{x} = \mathbf{x}_0 + t\mathbf{d}$. Geometrically this is exactly the line of intersection of the three planes: $\mathbf{x}_0$ is one point on the line and $\mathbf{d}$ is its direction. Recognising the parametric solution as a line ties this dot point back to vector equations of lines, and QCAA rewards stating the geometric interpretation alongside the algebra.

Reading the geometry of three planes

It helps to picture the configurations. Three planes meeting at a single point give a unique solution. Three planes sharing a common line give a one-parameter family. Three planes with no common point arise either when two are parallel and distinct, or when each pair meets in a line but the three lines are parallel (a triangular-prism arrangement). Coincident planes (all the same plane) give a two-parameter family. Matching the algebraic outcome to one of these pictures is the kind of justification an extended-response item asks for.