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Welcome to Engineering Mathematics 2

Welcome to Engineering Mathematics 2. We will cover 4 topics today 1 . Matrix Properties 2. Matrix Multiplication 3. Matrix Transpose 4. Matrix Questions. Lecture Recap.

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Welcome to Engineering Mathematics 2

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  1. Welcome to Engineering Mathematics 2 We will cover 4topics today 1. Matrix Properties 2. Matrix Multiplication 3. Matrix Transpose 4. Matrix Questions

  2. Lecture Recap We can also differentiate vector fields. We can find the divergence of a vector field (F) which is specified as Last week we learnt how to differentiate vectors. We looked at differentiation of scalar and vector fields. We can also find the curl of a vector field. This is calculated in the following manner If we differentiate a scalar field (φ) we obtain the gradient field

  3. Matrices We have seen that a rectangular array that obeys certain algebraic rules of operation is known as a matrix. A matrix is denoted using a capital letter. Where; 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. Hence, aij is the element in the ith row and the jth of the matrix A.

  4. Matrices Question Properties If A = [aij] and B = [bij] then Solve the equation A = B when A = B only if aij = bij for all elements Multiplication by a scalar k.A = [k.aij] Zero matrix: A = 0 only if [aij] = 0 Addition: A + B = [aij] + [bij] for all elements Subtraction: A - B = [aij] - [bij] for all elements

  5. Matrices Question If Then find A + B, B + A and A + 2B

  6. Matrix Properties An upper triangular matrix is defined as A row matrix is defined as A column matrix is defined as An lower triangular matrix is defined as A square matrix is defined as

  7. Matrix Properties The Identity or Unit matrix is defined as A diagonal matrix is defined as This can be written more concisely as [aij] = 0 if i ≠ j [aij] = 1 if i = j This matrix has the special property that This is also known as the leading diagonal. A diagonal matrix can be written more concisely as [aij] = 0 if i ≠ j We can also prove that I.A = A

  8. Matrix Multiplication The product is calculated as Not all matrices can be multiplied the two matrices must have the right shape. They must be conformable. The product of A and B only exists if the number of columns in A equals the number of rows in B. If the matrices are bigger then we get a matrix as the final result. If A is a 2 x 3 matrix and B is a 3 x 4 then A.B will be a 2 x 4 matrix. Consider the following row (A) and column (B) vectors. (2 x 3) (3 x 4) = 2 x 4

  9. Matrix Multiplication Consider the following matrix multiplication. If we use the first row of A as a row vector and the first column of B as a column vector then we can calculate their product as This is the result we obtained on the previous slide. If we wish to calculate the values of the remaining elements we need to use different row and column combinations. Element c11 is in the first row and first column of the solution matrix. Hence we find its result using the first row of A and the first column of B.

  10. Matrix Multiplication Question Find the elements c12 and c22 If we want to calculate element c21 we need to find the product of the second row of A and the first column of B, i.e. Thus

  11. Matrix Multiplication Question Find A postmultiplied by B (A.B) and A premultiplied by B (B.A) Let This is a special matrix known as the zero matrix. Hence, A.B ≠ B.A i.e. the matrix multiplication does not commute. The order of the multiplication is important. However, (A.B).C = A.(B.C). This is the associative law of multiplication

  12. Matrix Multiplication Question Find the set of linear equations for x, y & z represented by A.x = d If We have Thus In an earlier lecture we solved for x, y & z using determinants. We replaced the columns of matrix A with the column vector d to obtain 3 determinants. We then found the determinant of A and used these results to find x, y & z.

  13. Matrix Transpose Provided that if the sum A + B and the product A.B are defined then The transpose of any matrix is one in which the rows and the columns are interchanged. For example if A symmetric matrix is defined as Then An antisymmetric matrix is defined as i.e. I.e. row one has now become column one and row two is now column two.

  14. Matrix Transpose Question If Confirm that (A.B)T = BT.AT

  15. Matrix Section A Questions The value of the determinant Is? a) -96 b) -84 c) -72 d) 108

  16. Matrix Section A Questions If the matrices A, B and C are The value of the element C22 is? a) 0 b) 4 c) 6 d) 7

  17. Matrix Section A Questions If the matrices A, B and C are defined by Where x and y are constants. C is an antisymmetric matrix only when a) x = 1 for all values of y b) x = 3, y = -2 c) x = -1, y = -2 d) x = -1 for all values of y If x = -1, y = -2 then

  18. Matrix Section A Questions If the matrices A, B and C are defined by Where L and M are constants. C is a symmetric matrix for a) All values of L and M b) All values of L & M such that L = M c) L = -2, M = 0 d) L = 1, M = 1

  19. Conclusion Essential reading for next week HELM Workbook 7.1 Introduction to Matrices HELM Workbook 7.2 Matrix Multiplication We have covered 4topics today 1. Matrix Properties 2. Matrix Multiplication 3. Matrix Transpose 4. Matrix Questions

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