Use matrices beyond order two, including the determinant and inverse of a three by three matrix, to represent and solve systems of linear equations, and interpret unique, infinite and no-solution cases geometrically

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.