1. Matrices A matrix is a rectangular array of quantities (numbers, expressions or function), arranged in m rows and n columns. 131 41-2 -230 5 -2 1 2x 3y 4z

2. Special Matrices a11a12a13 a21a22a23 a31a32a33 a11a12a13 A11 A21 A31 100010001100010001 a1100 0a220 00a33 m = n Square matrix Column matrix Row matrix Identity matrix Diagonal matrix A = a ce b df A T = a b c d e f Matrix transpose

3. Scalar multiplication and Matrix addition If M = 1 23 4 56 3M = 3 69 12 15 18 a b c d  a  c b  d = a b c d e f  How about this??

4. Scalar products We can use matrices to represent vectors and use matrix multiplication to generate their scalar and vector products A = [a1, a2, a3], B = [b1, b2, b3] A.B = a1 a2 a3 = a1b1 + a2b2 + a3b3 b1 b2 b3 1212 12311231. =

5. Determinants of a Matrix If A = a11a12 a21a22 |A| = a11.a12 – a21.a12 Example If A = 24 -12 |A| = ?

6. A = a11a12a13 a21a22a23 a31a32a33 |A| = a11 a22 a23 – a12 a21 a23 + a13 a21 a22 a32 a33 a31 a33 a31 a32 Example A = 40-1 121 -365 Determinants of a Matrix

7. Properties of determinants |A| = |A T | Interchanging any two rows or any two columns of A changes the sign of |A| If we obtain B by multiplying one row or column of A by a constant, k then |B| = k|A| If two rows or columns of A is identical, then |A| = 0 If A square matrix and |A| = +1, it is orthogonal and proper. I |A| = -1, it is orthogonal and improper.

8. Matrix inversion The inverse of a square matrix A is A -1 AA -1 = A -1 A = I If an inverse exists, the matrix is said to be a nonsingular matrix, otherwise the matrix is called a singular matrix. Element of A -1 are a ij -1 where –a ij -1 = (-1) i+j |A ji | – |A|