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Further Pure 1

Further Pure 1. Matrices Introduction. Definitions. A matrix is just a rectangle of numbers. It’s a bit like a two-way table. You meet this concept in D1. The matrix below shows how many arcs exist between each node. Definitions Definitions. Here are more examples of matrices.

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Further Pure 1

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  1. Further Pure 1 Matrices Introduction

  2. Definitions • A matrix is just a rectangle of numbers. • It’s a bit like a two-way table. • You meet this concept in D1. • The matrix below shows how many arcs exist between each node.

  3. Definitions Definitions • Here are more examples of matrices. • You can see that each matrix is a different size. • The size (or order) of a matrix is given as rows × columns. • What is the order of each of the above matrices? = 2 × 2 = 2 × 5 = 4 × 1 = 3 × 3

  4. Definitions • Here are two special types of matrix that you need to be familiar with. • The identity matrix, I. • The zero matrix, O.

  5. Definitions • Matrices with the same number of rows and columns are known as square matrices. • Identity matrices are always square as the 1`s on the diagonal must run from corner to corner.

  6. Definitions • Two matrices are equal if: • They have the same order • Each element in one matrix is equal to the corresponding element in the other matrix.

  7. Using Matrices (+/-) • We can add and subtract matrices only if they have the same order. • All you do is add or subtract corresponding elements. • Why can you not add matrices with different orders?

  8. Using Matrices (×) • You can multiply a matrix by a number as illustrated in the example below. • All that has happened is each element has been multiplied by the number outside the matrix. • In general for any 2 × 2 matrix. • Remember that this will work for any matrix of any order.

  9. Problems • Explain why matrix addition is • Commutative, i.e. A + B = B + A • Associative, i.e. A + (B + C) = (A + B) + C • Addition in elements is both commutative and associative.

  10. Using Matrices (×) • Sometimes you can multiply two matrices together. • However not all matrices can be multiplied together. • Lets imagine 2 matrices called A and B. • If we want to calculate A × B then A must have the same number of columns as B has rows. • The sum you do is multiply each element in the 1st row of A by each element in the first column of B, then add together your answers. • You then do the same for all the row and column combinations. • On the next slide is a worked example.

  11. Using Matrices (×) • Let A be a 2 × 3 matrix and B be a 3 × 2 matrix. • So A × B is given by • Now take every element in the first row of A and multiply them by every element in the first column of B, adding your answers. • Now repeat with the 2nd row of A and 1st column of B. • Next 1st row of A and 2nd column of B. • Finally 2nd row of A and 2nd column of B.

  12. Using Matrices (×) • Try the numerical example below. • So A × B is given by. • Now take every element in the first row of A and multiply them by every element in the first column of B, adding your answers. • Now repeat with the 2nd of A and 1st column of B. • Next 1st row of A and 2nd column of B. • Finally 2nd row of A and 2nd column of B.

  13. Using Matrices (×) • What would happen if you found B × A. • So B × A is given by. • What do you notice about this answer compared to the last? • From these examples we can conclude that AB = BA • So matrix multiplication is not commutative

  14. Using Matrices (×) • If two matrices A and B have orders p × q and q × r respectively then A × B does exist, and will have order p × r. • Note: In this case B × A does not exist. p q q r p r × = q p r q ×

  15. Associative • Use the following matrices to show that matrix multiplication is associative. i.e. A(BC) = (AB)C

  16. Using Matrices (×) • Note that any matrix multiplied by the identity matrix is itself. • And any matrix multiplied by the zero matrix is the zero matrix.

  17. Questions • Try some of the following multiplications.

  18. Example • Below is a league table for the group stage of the world cup 2006. • The top 2 teams in each group progress through to the next round. • Use matrix multiplication to calculate the final points and hence state who progressed through to the next round.

  19. Solution • We can write the league table as a matrix. • Next we can add the matrix that represents the points awarded. • Its important to make sure that the correct points line up with the appropriate column. • Now we can multiply the two matrices together. • This shows us that England and Sweden progressed through to the next group.

  20. Summary • Matrix addition is Commutative • Matrix addition is Associative • Matrix multiplication is not Commutative • Matrix multiplication is Associative • AI = A • AO = O

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