Perform matrix operations, find determinants and inverses of 2x2 matrices, solve matrix equations, and apply transition matrices to model systems.

What this dot point is asking

This dot point tests matrix arithmetic, the inverse of a $2\times 2$ matrix, solving systems with matrices, and applying transition (Markov) matrices to real contexts.

Matrix operations

Addition and subtraction work entry by entry and require matrices of the same order. Scalar multiplication multiplies every entry by the scalar. Matrix multiplication is the key operation: the entry in row $i$, column $j$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$.

Determinant and inverse of a 2x2 matrix

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is

and the inverse, which exists only when $\det(A) \neq 0$, is

The inverse satisfies $A A^{-1} = A^{-1} A = I$, where $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is the identity.

Solving matrix equations

A system of linear equations can be written $AX = B$. If $A^{-1}$ exists, premultiply both sides:

Transition matrices

A transition matrix $T$ describes how proportions move between states each period. The state vector updates by $S_{n+1} = T S_n$, so $S_n = T^{\,n} S_0$. Columns of $T$ each sum to 1 because every member must end up in some state.

The system often settles to a steady state $S$ where $T S = S$; you can find it by computing $T^{\,n}$ for large $n$ on the calculator.