1.14 MatricesIB Maths: Applications and Interpretation HL: Revision notes

A matrix is a rectangular array of numbers. It has $m$ rows and $n$ columns, so its order is $m\times n$ (rows first). Each number is an element; $a_{ij}$ is the element in row $i$, column $j$.

• Equality: two matrices are equal if they have the same order and every pair of corresponding elements is equal.
• Addition and subtraction: only for matrices of the same order, element by element.
• Scalar multiplication: multiply every element by the scalar, e.g. $3\begin{pmatrix}1&-2\\ 0&4\end{pmatrix}=\begin{pmatrix}3&-6\\ 0&12\end{pmatrix}$. Your GDC can do all of this, and is the quickest way to handle large matrices.

$AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m\times n$ and $B$ is $n\times p$ then $AB$ is $m\times p$. Each element is a row of $A$ times a column of $B$. Example: $\begin{pmatrix}2&1\\ 3&4\end{pmatrix}\begin{pmatrix}1&0\\ -1&2\end{pmatrix}=\begin{pmatrix}1&2\\ -1&8\end{pmatrix}$, but reversing the order gives $\begin{pmatrix}2&1\\ 4&7\end{pmatrix}$.

• Associative: $A(BC)=(AB)C$.
• Distributive: $A(B+C)=AB+AC$.
• Not commutative: in general $AB\neq BA$. The identity matrix $I$ (ones on the main diagonal, zeros elsewhere) satisfies $AI=IA=A$. The zero matrix $0$ satisfies $A+0=A$ and $A0=0$.

For $A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ the determinant is $\det A=ad-bc$. If $\det A\neq0$ the inverse exists: $A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\ -c&a\end{pmatrix},\qquad AA^{-1}=A^{-1}A=I.$ Swap the diagonal elements, change the signs of the other two, then divide by the determinant. If $\det A=0$ the matrix is singular and has no inverse. Example: $A=\begin{pmatrix}2&1\\ 3&4\end{pmatrix}$ has $\det A=5$ and $A^{-1}=\frac15\begin{pmatrix}4&-1\\ -3&2\end{pmatrix}$. For $3\times3$ and larger matrices, find the determinant and inverse with your GDC.

A system of linear equations can be written $A\mathbf{x}=\mathbf{b}$, where $A$ holds the coefficients, $\mathbf{x}$ the unknowns and $\mathbf{b}$ the right-hand sides. In examinations $A$ is invertible, so multiply on the left by $A^{-1}$: $\mathbf{x}=A^{-1}\mathbf{b}.$ Example: $3s+2l=13.9$ and $2s+5l=21$ becomes $\begin{pmatrix}3&2\\ 2&5\end{pmatrix}\begin{pmatrix}s\\ l\end{pmatrix}=\begin{pmatrix}13.9\\ 21\end{pmatrix}$. Here $\det=11$ and the GDC gives $s=2.5$, $l=3.2$. The same method works for $3\times3$ systems with technology.

Matrices store and combine data in real problems.

• Costs and profits: a matrix of quantities times a vector of prices gives the total cost for each customer.
• Coding and decoding: write letters as numbers, group them as column vectors and multiply by a coding matrix $K$. To decode, multiply the coded vectors by $K^{-1}$; if $\det K=1$ the inverse has whole-number entries.
• Links: transition matrices in Markov chains (AHL 4.19) and phase portraits (AHL 5.17) are built on the same ideas. Always state what each row, column and answer represents, with units, and check that the orders allow the product.