Let a matrix organise a calculation

A matrix product uses rows of the first matrix and columns of the second. The inner dimensions must match. In general, changing the order changes the product.

For a square 2 × 2 matrix, a nonzero determinant allows an inverse and a unique solution to its linear system. A zero determinant signals dependent rows or columns, so there is no inverse.

\displaystyle \begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}

WORKED EXAMPLE

Solve $2x$ + y = 5 and x − y = 1.

Adding the equations gives 3x = 6, so x = 2.
Then y = 1.
The coefficient determinant is −3, consistent with a unique solution.

Assumed knowledge

Substitute into and solve simultaneous linear equations in two variables. Use signed arithmetic, the distributive law and ordered pairs.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Let a matrix organise a calculation

Solve 2x2x2x + y = 5 and x − y = 1.

Put the idea to work.

Solve $2x$ + y = 5 and x − y = 1.