1. Lesson 4 Review of Vectors and Matrices

2. Vectors A vector is normally expressed as or in terms of unit vectors likewise

3. Vectors (continued) Dot product Cross product

4. Matrices The general form for expressing a matrix operation is usually expressed as where A is the coefficient matrix consisting of n x m values, x is the unknown variable, and b is the RHS or load vector. Expanded in terms of the elements.

5. Matrices (continued) For example, if n = m = 3, we have Addition/Subtraction of matrices or

6. Matrices (continued) Multiplication of matrices If we multiply a 3 x 1 row vector, A, times a 1 x 3 column vector, B, we obtain On the other hand, B time A gives

7. Matrices (continued) Determinant of a matrix The determinant of a matrix is used in the 2 x 2 Jacobian matrix for transforming the shape function derivatives from  to x,y coordinates. The Jacobian is which can be written in the simpler form

8. Matrices (continued) The determinant, det A, is obtained by cross-multiplying and subtracting products, i.e., Hence, the determinant of the Jacobian matrix can be written as where N denotes the shape function and K the number of local node points

9. Matrices (continued) To find the inverse of a matrix (2 x 2)

10. Matrices (continued) Derivative of a matrix The derivative of a matrix is obtained by taking the derivative of each of its elements, i.e., The transpose of a row vector, A=[a 1 a 2 a 3 ] is