Revision:Matrices (Edexcel FP3)

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Revision:Matrices (Edexcel FP3)

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Matrices

Matrices are arrays of numbers, for example $\begin{pmatrix}1&2\\3&4\\ \end{pmatrix}$ .

1 Terminology
2 Addition and subtraction
3 Multiplication
4 Identity Matrix
5 Inverse of a Matrix
- 5.1 Finding the inverse of a 2×2 matrix
- 5.2 Finding the inverse of a 3×3 matrix
6 Comments

Terminology

Order

The order of a matrix denotes its size, i.e. how many rows and columns it contains. A matrix of order a×b has a rows and b columns. For instance:

$\begin{pmatrix} 1&-5\\ 2&9\\ -8&4\\ \end{pmatrix}$ is a matrix of order 3×2

Elements

Each number in a particular row and column in a matrix is called an element. For example in the matrix $\begin{pmatrix}8&11\\-1&23\\ \end{pmatrix}$ , the elements of the matrix are 8, 11, -1 and 23.

Square matrices

A square matrix is one where the two components of its order are equal - there are the same number of rows as columns. For instance: $\begin{pmatrix}8&11\\1&23\\ \end{pmatrix}$ and $\begin{pmatrix}9&-1&18\\2&21&13\\-6&17&-31\\ \end{pmatrix}$ are square matrices of order 2×2 and 3×3 respectively.

Transpose

The transpose of a matrix is formed by taking all the elements is the first column of the matrix and rewriting them as the first row of the transpose, second column to second row and so on. Vice versa also applies so that columns may be transposed to rows or rows to columns with the same result. For a matrix $\bf{A}$ the transpose is denoted $\bf{A^T}$ . For example:

$\bf{A} = \begin{pmatrix}1&2&3\\4&5&6\\7&8&9\\ \end{pmatrix} \\ \bf{A^T} = \begin{pmatrix}1&4&7\\2&5&8\\3&6&9\\ \end{pmatrix}$

Addition and subtraction

Any two matrices of the same order may be added or subtracted to/from each other. This is done by adding/subtracting the corresponding elements in each matrix. For example:

$\begin{pmatrix}8&2\\6&7 \end{pmatrix} + \begin{pmatrix}16&3\\0&1 \end{pmatrix} = \begin{pmatrix}24&5\\6&8 \end{pmatrix} \\ \begin{pmatrix}9&-3&0\\5&-11&7 \end{pmatrix} - \begin{pmatrix}5&1&10\\8&4&-3 \end{pmatrix} = \begin{pmatrix}4&-4&-10\\-3&-15&10 \end{pmatrix}$

Multiplication

You can multiply two matrices which fit the condition that the first matrix has the same number of columns as the second has rows.

Matrix multiplication is done by finding the sum of the products of the corresponding elements within the row of the first matrix and the column of the second that correspond to the row and column of the element being found. For instance to find the element in the first row and first column of the resulting matrix of a matrix multiplication, the first element in the first row of the first matrix is multiplied by the first element in the first column of the second matrix, and then this is added to the second element in the first row of the first matrix multiplied by the second element in the first column of the second matrix, and so on until the end of the row/column is reached. This process is repeated for each element. For example:

$\begin{pmatrix}7&5&-3\\8&1&0\\-5&9&2\end{pmatrix} \times \begin{pmatrix}2&-6&3\\4&6&1\\-9&0&2\end{pmatrix} \\ = \begin{pmatrix} (7 \times 2+5 \times 4+-3 \times -9)&(7 \times -6+5 \times 6+-3 \times 0)&(7 \times 3+5 \times 1+-3 \times 2)\\ (8 \times 2+1 \times 4+0 \times -9) &(8 \times -6+1 \times 6+0 \times 0) &(8 \times 3+1 \times 1+0 \times 2)\\ (-5 \times 2+9 \times 4+2 \times -9)&(-5 \times -6+9 \times 6+2 \times 0)&(-5 \times 3+9 \times 1+2 \times 2)\\ \end{pmatrix} = \begin{pmatrix} 61&-12&20\\ 20&-42&25\\ 18&84&-2\\ \end{pmatrix}$

Note that for two matrices $\bf{A}$ and $\bf{B}$ , generally $\bf{A} \times \bf{B} \neq \bf{B} \times \bf{A}$ , i.e. matrix multiplication is normally non-commutative.

Identity Matrix

An identity matrix is a square matrix whereby each of the elements in the leading diagonal (top left to bottom right) are ones and all others elements are zeroes. It is denoted $\bf{I}$ . For instance the 2×2 identity matrix is:

$I_2 = \begin{pmatrix}1&0\\0&1\end{pmatrix}$

In general identity matrices have the special property, that for a matrix $\bf{A}$ and identity matrix $\bf{I}$ of the same order, $\bf{AI} = \bf{IA} = \bf{A}$ .

Inverse of a Matrix

For two square matrices $\bf{A}$ and $\bf{B}$ , where $\bf{AB} = \bf{I}$ ,

$\bf{A}$ is said to be the inverse of $\bf{B}$ and vice versa.

This is written $\bf{A}=\bf{B^{-1}$ and $\bf{B}=\bf{A^{-1}$ .

Finding the inverse of a 2×2 matrix

For a matrix $\bf{A}= \begin{pmatrix}p&q\\r&s\end{pmatrix}$

First the determinant of $\bf{A}$ is found:

$det \bf{A}=\begin{vmatrix}p&q\\r&s\end{vmatrix}= ps-qr$

Then the elements of $\bf{A}$ are rearranged and their signs altered as follows:

$\begin{pmatrix}s&-q\\-r&p\end{pmatrix}$

Then the inverse is found by dividing the rearranged matrix by the determinant (a scalar constant):

$\bf{A^{-1}}= \frac{1}{ps-qr}\begin{pmatrix}s&-q\\-r&p\end{pmatrix}$

In the case where ps-qr=0 , there is no inverse and so the matrix is called singular.

Finding the inverse of a 3×3 matrix

If you take

$\mathbf{M} = \begin{pmatrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \\ c_1&c_2&c_3 \end{pmatrix}$

and then let

$A_1 = \det \begin{pmatrix} b_2&b_3 \\ c_2&c_3 \end{pmatrix} ,$ $A_2 = - \det \begin{pmatrix} b_1&b_3 \\ c_1&c_3 \end{pmatrix} ,$ $A_3 = \det \begin{pmatrix} b_1&b_2 \\ c_1&c_2 \end{pmatrix} ,$ $B_1 = - \det \begin{pmatrix} a_2&a_3 \\ c_2&c_3 \end{pmatrix}$ etc. (the cofactors)