Introduction to Matrices (Edexcel IAL Further Maths)

Spec point — 5.1

Introduction to Matrices

What is a matrix?

• A matrix is a rectangular grid (array) of elements (numbers or letters) arranged in rows and columns

  • The plural of matrix is matrices

• The order (dimensions) of a matrix is its number of rows × number of columns

  • a 2 × 1 matrix is $( \begin{gathered} a \\ b \end{gathered} )$

    • this is also called a column matrix or a column vector

  • a 2 × 2 matrix is $( \begin{gathered} a b \\ c d \end{gathered} )$

    • this is called a square matrix

• A bold capital letter is often used to represent a matrix

  • $A = ( \begin{gathered} 5 2 \\ 0 4 \end{gathered} )$, $B = ( \begin{gathered} 4 \\ 3 \end{gathered} )$

• 2D coordinates can be written as a column matrix

  • The point $( 3 , 5 )$ is $( \begin{gathered} 3 \\ 5 \end{gathered} )$

• You can use subscript notation to refer to elements in a matrix

  • Matrix $A$ is $A=(a_{i,j})$ where $i=1,2,3,...,m$ and $j=1,2,3,...,n$

    • $a_{i,j}$ refers to the element in row $i$, column $j$
    • The order of $A$ is $m\times n$ (rows × columns)

What type of matrices do I need to know?

• A column matrix (or column vector) is a matrix with a single column

  • Order $m\times1$

• A row matrix is a matrix with a single row

  • Order $1\times n$

• A square matrix is one in which the number of rows is equal to the number of columns

  • Order $n\times n$

• Two matrices are equal when they are of the same order and their corresponding elements are equal

  • $a_{i,j}=b_{i,j}$ for all elements

• A zero matrix, $0$, is a matrix in which all the elements are zero

  • For example, the 2 × 2 zero matrix is $0 = ( \begin{gathered} 0 0 \\ 0 0 \end{gathered} )$

• An identity matrix, $I$, is a square matrix in which all elements along the leading diagonal (top-left to bottom right) are 1

  • The rest of the elements are zero
  • For example, the 2 × 2 identity matrix is $I = ( \begin{gathered} 1 0 \\ 0 1 \end{gathered} )$
  • The notation $I_{n}$ can be used to specify the $n\times n$ identity matrix

Basic Operations with Matrices

Spec point — 5.3

How do I multiply a matrix by a scalar?

• To multiply any matrix by a scalar (a number), multiply each element by that scalar

  • If $A = ( \begin{gathered} 5 2 \\ 0 4 \end{gathered} )$ then $2 A = 2 ( \begin{gathered} 5 2 \\ 0 4 \end{gathered} ) = ( \begin{gathered} 2 \times 5 & 2 \times 2 \\ 2 \times 0 & 2 \times 4 \end{gathered} ) = ( \begin{gathered} 10 4 \\ 0 8 \end{gathered} )$

• Multiplying by a negative scalar changes the sign of each element in the matrix

• Lower case letters often refer to scalar multiples

  • $kA$ is the matrix $A$ multiplied by the scalar $k$

How do I add and subtract matrices?

• Two matrices of the same order can be added (or subtracted) by adding (or subtracting) corresponding elements

  • The answer is a matrix of the same order
  • For example, $( \begin{gathered} 1 2 \\ - 3 & 0 \\ 2 1 \end{gathered} ) + ( \begin{gathered} 3 0 \\ 3 - 1 \\ 4 1 \end{gathered} ) = ( \begin{gathered} 1 + 3 & 2 + 0 \\ - 3 + 3 & 0 - 1 \\ 2 + 4 & 1 + 1 \end{gathered} ) = ( \begin{gathered} 4 2 \\ 0 - 1 \\ 6 2 \end{gathered} )$

What properties of matrix addition do I need to know?

• $A+B=B+A$

  • Matrix addition is commutative

    • You can swap the order

  • Matrix subtraction is not commutative

    • $A-B\neq B-A$

• $A-B=A+(-B)$

  • Subtraction is the same as adding a negative

• $A+(B+C)=(A+B)+C$

  • Matrix addition is associative

    • To add three matrices, you can start with the first two, or the last two

  • Matrix subtraction is not associative

    • $A-(B-C)\neq(A-B)-C$
    • Try expanding the brackets to see

• $A+0=A$

  • Adding the zero matrix has no effect

• $0-A=-A$

Worked Example Consider the matrices $A = ( \begin{gathered} - 4 & 2 \\ 7 3 \\ 1 - 5 \end{gathered} )$, $B = ( \begin{gathered} 2 6 \\ 5 - 9 \\ - 2 & - 3 \end{gathered} )$.

(a) Find $A+B$.

The matrices have the same order (dimensions) Corresponding elements can be added

$A + B = ( \begin{gathered} - 4 + 2 & 2 + 6 \\ 7 + 5 & 3 + ( - 9 ) \\ 1 + ( - 2 ) & - 5 + ( - 3 ) \end{gathered} )$

$A + B = ( \begin{gathered} - 2 & 8 \\ 12 - 6 \\ - 1 & - 8 \end{gathered} )$

(b) Find $-10B$.

Multiply each element by -10

• *$\begin{gathered} - 10 B & = - 10 ( \begin{gathered} 2 6 \\ 5 - 9 \\ - 2 & - 3 \end{gathered} ) \\ = ( \begin{gathered} - 10 \times 2 & - 10 \times 6 \\ - 10 \times 5 & - 10 \times ( - 9 ) \\ - 10 \times ( - 2 ) & - 10 \times ( - 3 ) \end{gathered} ) \end{gathered}$

$\begin{gathered} - 10 B & = ( \begin{gathered} - 20 & - 60 \\ - 50 & 90 \\ 20 30 \end{gathered} ) \end{gathered}$