Introduction to Matrices (Edexcel IGCSE Maths)

Spec point — 5A

Introduction to matrices

What is a matrix?

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

  • They are used a lot in computer / data science
  • The plural of "matrix" is "matrices"

• The order of a matrix is its "number of rows" × "number of columns"

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

    • this is also called a column matrix or a column vector
    • this could be used to represent the point with coordinates$( a , b )$

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

    • this is called a square matrix (same number of rows and columns)

  • a 2 × 3 matrix is $( \begin{gathered} a b \\ d e \end{gathered} \begin{gathered} c \\ f \end{gathered} )$

    • this could be used to represent the vertices of a triangle with coordinates $( a , d )$, $( b , e )$ and $( c , f )$

  • a 3 × 3 matrix is $( \begin{gathered} a b \\ d e \\ g h \end{gathered} \begin{gathered} c \\ f \\ i \end{gathered} )$

    • this is also a square matrix (same number of rows and columns)

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

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

Exam Hint The biggest matrices you will need to deal with in this course are 3 × 3 matrices.

The identity and null matrices

Spec point — 5B

What is the identity (or unit) matrix?

• The identity matrix, $I$, is a square matrix with 1s along the diagonal from top-left to bottom-right and zeros everywhere else

  • The 2×2 identity matrix is $I = ( \begin{gathered} 1 0 \\ 0 1 \end{gathered} )$
  • The 3×3 identity matrix is $I = ( \begin{gathered} 1 0 0 \\ 0 1 0 \\ 0 0 1 \end{gathered} )$

• It is also sometimes known as the unit matrix

  • For matrices, it fulfils a role similar to the number 1 for regular numbers

What is the null (or zero) matrix?

• The null matrix is a matrix with all its elements equal to zero

  • The 2×2 null matrix is $( \begin{gathered} 0 0 \\ 0 0 \end{gathered} )$
  • The 3×3 null matrix is $( \begin{gathered} 0 0 0 \\ 0 0 0 \\ 0 0 0 \end{gathered} )$

• Unlike the identity matrix, the null matrix does not have to be square

  • E.g., the 2×3 null matrix is $( \begin{gathered} 0 0 \\ 0 0 \end{gathered} \begin{gathered} 0 \\ 0 \end{gathered} )$

• It is also sometimes known as the zero matrix

  • For matrices, it fulfils a role similar to the number 0 (zero) for regular numbers
  • You may sometimes see a bold zero, $0$, used to represent the null matrix