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. Lecture Recap The product of a row and column vector is calculated as Last lecture we learnt how to multiply matrices. Remember that not all matrices can be multiplied, the two matrices must have the right shape. They must be conformable. If A is an m x n matrix and B is a p x q matrix then A.B will be a m x q matrix provided that n = p If we wish to calculate the product of a larger matrix. Each element is calculated following the procedure above. However, we must use the appropriate row and column of the matrices A and B respectively. (m x n) (p x q) = m x q

3. Determinants Now consider these simultaneous equations. Today we will look at the definition of a determinant and its properties. In particular we will be looking at 2nd order and 3rd order determinants. What are x and y? Consider the following simultaneous equations. Provided that a.d – b.c ≠ 0 What are x and y? We will look at this generic solution again later. x = 1, y = -3

4. 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 Ω is a 2 x 2 matrix. = 0 x 0 – (-1) x 1 = 1

5. What is Determinants Recall that if Then we can solve for the unknowns using Hence, express x and y in terms of these determinants Provided that a.d – b.c ≠ 0 This is called Cramer’s rule

6. Determinants Solve the following simultaneous equations using Cramer’s rule.

7. Determinants Evaluate the following simultaneous equations using Cramer’s rule. What has gone wrong?

8. 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?

9. Properties of 2nd Order 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.

10. Properties of 2nd Order 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 all the columns or rows are identical the value of the determinant is zero.

11. Properties of 2nd Order Determinants 5) Multiplying all the elements of one row or column by a constant (λ) multiplies the value of the determinant by λ. 6) The addition rule

12. Properties of 2nd Order Determinants 7) A multiple of one row or column can be added to the other without changing the value of the determinant. Similarly,

13. 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.

14. 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

15. Third Order Determinants How do we determine the cofactors? How do we determine the minors? First we must defined the minor of an element. The minor of the element aij is found by removing both the row and the column containing aij and then finding the determinant of the remainder. For example, taking the following matrix

16. Third Order Determinants The cofactor of an element is equal to either the positive or negative of the minor. 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 matrix. Thus, the cofactors of the elements a11, a13, a22, a31 & a33 are positive i.e. the cofactor of a11 is Whereas, the cofactors of the elements a12, a21, a23 & a32 are negative i.e. the cofactor of a12 is This matrix defines the signs of the cofactors.

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

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

19. 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.

20. 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,

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

22. Third Order Determinants The value of the determinant Is? a) -96 b) -84 c) -72 d) 108

23. Third Order Determinants Properties 3rd order determinants have the same properties as 2nd order determinants; however, be careful with property 2. 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. Conclusion Essential reading for next week HELM Workbook 7.3 Determinants We have covered 4 topics today 1. Matrix Definition and Notation2. Determinants 3. Properties of Determinants 4. 3rd Order Determinants