Perform matrix addition, scalar multiplication and matrix multiplication, find the determinant and inverse of a 2x2 matrix, and use the inverse to solve a system of simultaneous linear equations

What this dot point is asking

VCAA wants you to do arithmetic with matrices and to use the inverse of a $2\times 2$ matrix to solve a pair of simultaneous linear equations. You must know when two matrices can be added or multiplied (conformability), perform the row-by-column product, compute a determinant and inverse by hand, and set up a system in the form $AX = B$. This is the foundation of the Matrices module and is examined on both papers.

Addition and scalar multiplication

To add or subtract matrices of the same order, add or subtract the corresponding entries. To multiply a matrix by a scalar $k$, multiply every entry by $k$. For example,

Matrix multiplication

The product $AB$ is found by the row-by-column rule: the entry in row $i$, column $j$ of $AB$ is the sum of the products of row $i$ of $A$ with column $j$ of $B$. If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$. Matrix multiplication is not commutative: in general $AB \ne BA$.

The identity and the inverse

The $2\times 2$ identity matrix is $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, and $AI = IA = A$. The inverse $A^{-1}$ satisfies $A A^{-1} = A^{-1} A = I$. A matrix with determinant zero is singular and has no inverse.

Solving simultaneous equations

A pair of linear equations can be written as a matrix equation $AX = B$, where $A$ holds the coefficients, $X$ the unknowns and $B$ the constants. Multiplying both sides on the left by $A^{-1}$ gives

Why this matters for the exams

Matrix arithmetic is assumed knowledge for everything else in the module, including transition matrices and matrix models of populations. Exam 1 typically tests order, a small product, or a determinant by hand; Exam 2 may set up a system and expect you to solve it with the calculator using $X = A^{-1}B$, then interpret the answer in context.