Welcome to Engineering Mathematics 2

1. Welcome to Engineering Mathematics 2 We will cover 4 topics today 1. Matrix Definition and Notation2. Determinants 3. Properties of Determinants 4. 3rd Order Determinants

2. Matrix Notation A rectangular array that obeys certain algebraic rules of operation is known as a matrix. A general m x n matrix is one with m rows and n columns. Hence a general matrix can be represented in the following way. This is an example of a matrix. A matrix is denoted using a capital letter. Where; This is a matrix with two rows and three columns. This matrix is said to be of order 2 x 3. Or, a 2 x 3 matrix. Hence, aij is the element in the ith row and the jth of the matrix A.

3. Matrix Notation Question The elements of a matrix are given by Where i = 1, 2, 3 & j = 1, 2 ,3. Write out the matrix in full.

4. Matrix Notation Question The elements of a matrix are given by Where i = 1, 2 & j = 1 → 6. Write out the matrix in full.

5. Determinants Now consider these simultaneous equations. Consider the following simultaneous equations. What are x and y? What are x and y? x = 1, y = -3 Provided that a.d – b.c ≠ 0

6. Determinants Consider a 2 x 2 matrix Ω Evaluate The determinant of this matrix is defined as = 9 x (-1) – (-1) x 7 = -2 Evaluate This is called a second order determinant because this is a 2 x 2 matrix. = 0 x 0 – (-1) x 1 = 1

7. Determinants What is Recall that if Then Hence, express x and y in terms of these determinants Provided that a.d – b.c ≠ 0

8. Determinants Solve the following simultaneous equations using determinants.

9. Determinants Evaluate the following simultaneous equations using determinants What has gone wrong?

10. Determinants If a matrix has a determinant that is equal to zero then the matrix is said to be Singular. A singular matrix does not have an inverse. Using only 1s and 0s. Can you find 10 2x2 singular matrices?

11. Properties of Determinants 1) The value of the determinant is unaltered if it is transposed (i.e. the rows and columns are interchanged). 2) Interchanging the rows or the columns changes the sign of the determinant.

12. Properties of Determinants 3) If all the elements of either one row or one column are all zero then the value of the determinant is zero. 4) If the columns or rows are identical the value of the determinant is zero.

13. Properties of Determinants 5) Multiplying all the elements of a row or column by a constant (λ) multiplies the value of the determinant by λ. 6) The addition rule

14. Properties of Determinants 7) A multiple of one row or column can be added to the other without changing the value of the determinant. Similarly,

15. Third Order Determinants For example, if we solve by expanding along the top row we get: A 3rd order determinant is the determinant of a 3rd order matrix i.e. What is the determinant of A? The solution involves an expansion along an arbitrary column or row.

16. Third Order Determinants A 3rd order determinant expanded along the top row can be written as Likewise, the cofactor of a12 is Where Cij stands for the cofactor of aij. Finally, the cofactor of a13 is The cofactor of a11 is

17. Third Order Determinants How do we determine the cofactors? How do we determine the cofactors? The cofactor of the element aij is found be removing both the row and the column containing aij and then finding the determinant of the remainder. For example, taking the following matrix Ignore the sign for a minute.

18. Third Order Determinants Thus, the cofactors of the elements a11, a13, a22, a31 & a33 are positive i.e. the cofactor of a11 is On the previous slide you will have noticed that the cofactor of a12 was negative. There is a convention that defines the sign of the cofactor. The sign of the cofactor depends on the place of the relevant element in the matrix. Consider the following matrices. Whereas, the cofactors of the elements a12, a21, a23 & a32 are negative i.e. the cofactor of a12 is NB. If you do not multiply by ± 1, then you are only calculating the minor of the element.

19. Third Order Determinants Using the following matrix, determine the cofactors C22, C23, C32 & C33

20. Third Order Determinants A third order determinant can be found by expanding along a row or a column. If we expand along a row (i) the determinant is found by (‘i’ can be equal to 1, 2 or 3). If we expand along a column (j) the determinant is found by (‘j’ can be equal to 1, 2 or 3). When calculating the determinant, it does not matter which row or column you expand along. The result will be the same.

21. Third Order Determinants Question The matrix A is defined by Find the determinant by expanding about the easiest row. The easiest row is the 3rd row; hence,

22. Third Order Determinants Question The matrix A is defined by Find the determinant by expanding about the second column.

23. Third Order Determinants Properties 3rd order determinants have the same properties as 2nd order determinants with one exception. This is Property 2 Interchanging any two adjacent rows (or columns) changes the sign of the determinant i.e. The same result is true when interchanging columns.

24. Third Order Determinants Question The matrix Ω is defined by • Calculate the minors of • ω11 • ω12 • ω13 • Calculate the cofactors of • ω21 • ω22 • ω23

25. Conclusion Essential reading for next week HELM Workbook 7.1 An Introduction to Matrices HELM Workbook 7.2 Determinants We have covered 4 topics today 1. Matrix Definition and Notation2. Determinants 3. Properties of Determinants 4. 3rd Order Determinants