Introduction to Matrices (Edexcel A Level Further Maths: Core Pure)

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15 November 2024

Introduction to Matrices

Matrices are a useful way to represent and manipulate data in order to model situations. The elements in a matrix can represent data, equations or systems and have many real-life applications.

What are matrices?

A matrix is a rectangular array of elements (numerical or algebraic) that are arranged in rows and columns
The order of a matrix is defined by the number of rows and columns that it has
- The order of a matrix with $m$ rows and $n$ columns is $m \times n$
A matrix $A$ can be defined by $A = (a_{i j})$ where $i = 1, 2, 3, . . ., m$ and $j = 1, 2, 3, . . ., n$ and $a_{i j}$ refers to the element in row $i$ , column $j$

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What type of matrices are there?

A column matrix (or column vector) is a matrix with a single column, $n = 1$
A row matrix is a matrix with a single row, $m = 1$
A square matrix is one in which the number of rows is equal to the number of columns, $m = n$
Two matrices are equal when they are of the same order and their corresponding elements are equal, i.e. $a_{i j} = b_{i j}$ for all elements
A zero matrix, $O$ , is a matrix in which all the elements are $0$ , e.g. $O = (\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix})$
The identity matrix, $I$ , is a square matrix in which all elements along the leading diagonal are $1$ and the rest are $0$ , e.g. $I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$

What is the transpose of a matrix?

The transpose of matrix A is denoted as A^T
The transpose matrix is formed by interchanging the rows and columns

$A^{T} = (\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix}) A^{T} = (\begin{matrix} a_{11} & a_{21} \\ a_{12} & a_{22} \\ a_{13} & a_{23} \end{matrix})$

Examiner Tip

Make sure that you know how to enter and store a matrix on your calculator

Worked example

Let the matrix $A = (\begin{matrix} 5 & - 3 & 7 \\ - 1 & 2 & 4 \end{matrix})$

Write down the order of

A

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State the value of

a_{2, 3}

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Basic Operations with Matrices

Just as with ordinary numbers, matrices can be added together and subtracted from one another, provided that they meet certain conditions.

How is addition and subtraction performed with matrices?

Two matrices of the same order can be added or subtracted
Only corresponding elements of the two matrices are added or subtracted
- $A \pm B = (a_{i j}) \pm (b_{i j}) = (a_{i j} \pm b_{i j})$
The resultant matrix is of the same order as the original matrices being added or subtracted

What are the properties of matrix addition and subtraction?

$A + B = B + A$ (commutative)
$A + (B + C) = (A + B) + C$ (associative)
$A + O = A$
$O - A = - A$
$A - B = A + (- B)$

How do I multiply a matrix by a scalar?

Multiply each element in the matrix by the scalar value
- $k A = (k a_{i j})$
The resultant matrix is of the same order as the original matrix
Multiplication by a negative scalar changes the sign of each element in the matrix

Worked example

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

Find

A + B

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Find

A - B

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Multiplying Matrices

Can I always multiply a matrix by another matrix?

Not always - only if the dimensions of the matrices allow it
If A has order $m \times n$ and B has order $q \times p$ then you the matrix AB exists only if $n = q$
The order of the matrix AB will be $m \times p$
It is possible for AB to exist but BA not exist and vice versa
AB and BA both will exist if they are both square matrices of the same order
- This means the dimensions are the same $n \times n$

How do I multiply a matrix by another matrix?

To multiply a matrix by another matrix, the number of columns in the first matrix must be equal to the number of rows in the second matrix
If the order of the first matrix is $m \times n$ and the order of the second matrix is $n \times p$ , then the order of the resultant matrix will be $m \times p$
The product of two matrices is found by multiplying the corresponding elements in the row of the first matrix with the corresponding elements in the column of the second matrix and finding the sum to place in the resultant matrix

E.g. If $A = [\begin{matrix} a & b & c \\ d & e & f \end{matrix}]$ , $B = [\begin{matrix} g & h \\ i & j \\ k & l \end{matrix}]$

then $A B = [\begin{matrix} (a g + b i + c k) & (a h + b j + c l) \\ (d g + e i + f k) & (d h + e j + f l) \end{matrix}]$
then $B A = [\begin{matrix} (g a + h d) & (g b + h e) & (g c + h f) \\ (i a + j d) & (i b + j e) & (i c + j f) \\ (k a + l d) & (k b + l e) & (k c + l f) \end{matrix}]$

How do I square an expression involving matrices?

If an expression involving matrices is squared then you are multiplying the expression by itself, so write it out in bracket form first, e.g. ${(A + B)}^{2} = (A + B) (A + B)$
- remember, the regular rules of algebra do not apply here and you cannot expand these brackets, instead, add together the matrices inside the brackets and then multiply the matrices together

What are the properties of matrix multiplication?

$A B \neq B A$ (non-commutative)
$A (B C) = (A B) C$ (associative)
$A (B + C) = A B + A C$ (distributive)
$(A + B) C = A C + B C$ (distributive)
$A I = I A = A$ (identity law)
$A O = O A = O$ , where $O$ is a zero matrix
Powers of square matrices: $A^{2} = A A, A^{3} = A A A$ etc.

Worked example

Consider the matrices $A = [\begin{matrix} 4 & 2 & - 5 \\ - 3 & 8 & 1 \\ - 1 & - 2 & 2 \end{matrix}]$ and $B = [\begin{matrix} 5 & 1 \\ - 2 & 5 \\ 9 & 7 \end{matrix}]$ .

Find