1. Chapter 2 Solving Linear Systems Matrix Definitions –Matrix--- Rectangular array/ block of numbers. –The size/order/dimension of a matrix: (The numbers of ROWS) by(x) (the numbers of COLUMNS)

2. –ELEMENTS: individual numbers of matrix –a ij --- an element of ROW i and COLUMN j –SQURE matrix The numbers of ROWS = the numbers of COLUMNS –IDENTITY matrix: symbol---I –TRANSPOSED matrix: Rows and columns of a matrix are switched –

3. Matrix Operations –Addition Two same size matrices can be added. C=A+B=B+A

4. –Multiplication Multiplication of a Matrix by a Scalar –A=kA –Example Multiplication of 2 Matrices –Two Matrix can be multiplied if and only if--- The NUMBER OF COLUMNS OF THE FIRST MATRIX = The NUMBER OF ROWS OF THE SECOND MATRIX –The Size of the resultant matrix --- the NUMBER OF ROWS OF THE FIRST MATRIX by the NUMBER OF COLUMNS OF THE SECOND MATRIX

5. Example First Matrix Second Matrix Multipication Size Possible? A B AB (a ）（ 2x2) (2x2) YES (2x2) (b ）（ 3x3) (3x2) YES (3x2) (c ）（ 3x3) (2x3) NO (d ）（ 5x5) (5x1) YES (5x1)

6. Notice that: –AB exists and so does BA with BA being (2x2) –AB exists, BA does not exist as a (3x2) cannot be multiplied into a (3x3) –AB does not exist, It’s possible that BA exists How to calculate the elements of C=AB –Example

8. –A---m x n matrix I=identity matrix »I A = A »A I = A

9. –Matrix Inversion Only Square matrices have the inverse but not all square matrices have inverses. Scalar number: The inverse of matrix A is denoted by A -1 The size of A -1 is the same as A and A A -1 = I = A -1 A Any Matrix times its own inverse is just the appropriately sized identity matrix

10. –Matrix Equality Two matrices are said to be equal if –They are same size –Corresponding elements in the two matrices are the same

11. Break-Even Model in Matrix Algebra terms – Break-even model in linear equations 1 TR + 0 TC – 20q = 0 0 TR + 1 TC – 25q = 500 1 TR – 1 TC + 0q = 0 –Let

12. –Ax=b  A -1 Ax= A -1 b  I x= A -1 b  x= A -1 b –Example

13. –Modelling Steps Set up the system of linear equations Decide upon an order in which to express the unknowns The unknowns on the LHS of the equations Identify the following 3 matrices –A: Square matrix of coefficients relating to the unknowns –x: the matrix of unknows –b: the matrix of RHS constants Find matrix inverse A -1 of A Perform the matrix multiplication A -1 b Use the matrix equality rule to find the elements of x Give the business interpretation of x