Introduction to Matrices (Edexcel International AS Further Maths: Further Pure 1): Revision Note

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 $\left(\begin{array}{c}a\\ b\end{array}\right)$

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

  • a 2 × 2 matrix is $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

    • this is called a square matrix

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

  •  $\mathbf{A}=\left(\begin{array}{cc}5& 2\\ 0& 4\end{array}\right)$, $\mathbf{B}=\left(\begin{array}{c}4\\ 3\end{array}\right)$

• 2D coordinates can be written as a column matrix

  • The point $\left(3,5\right)$ is $\left(\begin{array}{c}3\\ 5\end{array}\right)$ 

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

  • Matrix $\mathbf{A}$ is $\mathbf{A}=\left(a_{i,j}\right)$ 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 $\mathbf{A}$ 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 $m\times 1$

• 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, $\mathbf{0}$, is a matrix in which all the elements are zero

  • For example, the 2 × 2 zero matrix is $\mathbf{0}=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$

• An identity matrix, $\mathbf{I}$, 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  $\mathbf{I}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

  • The notation ${\mathbf{I}}_n$ can be used to specify the $n\times n$ identity matrix

Basic Operations with Matrices

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 $\mathbf{A}=\left(\begin{array}{cc}5& 2\\ 0& 4\end{array}\right)$ then $2\mathbf{A}=2\left(\begin{array}{cc}5& 2\\ 0& 4\end{array}\right)=\left(\begin{array}{cc}2\times 5& 2\times 2\\ 2\times 0& 2\times 4\end{array}\right)=\left(\begin{array}{cc}10& 4\\ 0& 8\end{array}\right)$

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

• Lower case letters often refer to scalar multiples

  • $k\mathbf{A}$ is the matrix $\mathbf{A}$ 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, $\left(\begin{array}{cc}1& 2\\ -3& 0\\ 2& 1\end{array}\right)+\left(\begin{array}{cc}3& 0\\ 3& -1\\ 4& 1\end{array}\right)=\left(\begin{array}{cc}1+3& 2+0\\ -3+3& 0-1\\ 2+4& 1+1\end{array}\right)=\left(\begin{array}{cc}4& 2\\ 0& -1\\ 6& 2\end{array}\right)$

What properties of matrix addition do I need to know?

• $\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}$

  • Matrix addition is commutative

    • You can swap the order

  • Matrix subtraction is not commutative

    • $\mathbf{A}-\mathbf{B}\ne \mathbf{B}-\mathbf{A}$

• $\mathbf{A}-\mathbf{B}=\mathbf{A}+(-\mathbf{B})$

  • Subtraction is the same as adding a negative

• $\mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C}$

  • Matrix addition is associative

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

  • Matrix subtraction is not associative

    • $\mathbf{A}-(\mathbf{B}-\mathbf{C})\ne (\mathbf{A}-\mathbf{B})-\mathbf{C}$

    • Try expanding the brackets to see

• $\mathbf{A}+\mathbf{0}=\mathbf{A}$

  • Adding the zero matrix has no effect

• $\mathbf{0}-\mathbf{A}=-\mathbf{A}$