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•Addition of Matrices •Subtraction of Matrices •Scalar Multiplication of Matrix •Matrix Multiplication •Determinant of 2 x 2 matrix •Inverse Matrix of 2 x 2 matrix •Solving SLE using Matrix Method Matrix – rectangular arrangement of variables or constants into horizontal rows and vertical columns. 4 columns Example: 3 2 1 0 4 0 2 1 7 13 2 5 This would be a 3 x 4 Order – number of rows by number of columns 3 rows •Row Matrix – only one row •Column Matrix – only one column •Square Matrix – same number of rows as columns •Equal matrices – same order and each entry of one matrix is equal to the corresponding entry of the other matrix. Addition and Subtraction of Matrices To add or subtract matrices, they must have the same order – same number of rows and columns – because you just add or subtract corresponding entries. Example: 2 3 7 5 6 0 9 5 12 7 3 8 7 0 7 9 7 2 5 3 6 9 7 5 3 12 8 16 2 4 Multiplication of Matrices Scalar multiplication – multiply the entire matrix by a number Example: 2 9 3 0 1 5 12 6 27 0 3 15 36 Multiplication of Matrices Matrix multiplication – two matrices can only be multiplied if the number of columns in the first equals the number of rows in the second. 2x3 could be multiplied with a 3x4 could not multiply 3x4 and 3x4 The order of the product matrix (what you get after you multiply) will be the number of rows from the first and the number of column from the second. When you multiply the 2x3 and the 3x4, the product will be a 2x4 Matrix multiplication – to multiply two matrices, you multiply each row in the first by each column in the second. 3 2 Example: 1 2 0 3 5 2 0 4 1 1 2x3 and 3x2…can multiply and the product will be a 2x2 (1)(3) (2)(0) (0)(1) (3)(3) (5)(0) (2)(1) (1)(2) (2)(4) (0)(1) (3)(2) (5)(4) (2)(1) 3 10 7 12 Example : A motor manufacturer, with three separate factories, makes two types of car -one called “standard” and the other called “luxury”. In order to manufacture each type of car, he needs a certain number of units of material and a certain number of units of labour each unit representing £300. A table of data to represent this information could be Type Materials Labour Standard 12 15 Luxury 16 20 The manufacturer receives an order from another country to supply 400 standard cars and 900 luxury cars. He distributes the export order as follows: Location Standard Luxury Factory A 100 400 Factory B 200 200 Factory C 100 300 The number of units of material and labour needed to complete the order may be given by the following table: Location Materials Labour Factory A 100 × 12 + 400 × 16 100 × 15 + 400 × 20 Factory B 200 × 12 + 200 × 16 200 × 15 + 200 × 20 Factory C 100 × 12 + 300 × 16 100 × 15 + 300 × 20 Determinants Every square matrix has a number associated with it called a determinant. Second – order determinant denoted by: a b a b det or c d c d = ad - bc Product of the diagonal going down minus the product of the diagonal going up Examples: 3 10 Find det 4 5 = (3)(-5) – (10)(4) = -15 – 40 = -55 1 4 Find 3 0 = (1)(0) – (-4)(3) = 0 – -12 = 12 Identity and Inverse Matrices Identity matrix is a square matrix that when multiplied by another matrix, the product equals that same matrix. Identity matrix : 1 0 0 1 , 1 1 0 0 0 1 0 , 0 0 0 0 1 0 0 0 0 1 0 0 , etc 0 1 0 0 0 1 Identity Matrix has 1 for each element on the main diagonal and 0 everywhere else. matrix times inverse = identity matrix A A I 1 Not every matrix has an inverse. Requirements to have an Inverse • The matrix must be square (same number of rows and columns). • The determinant of the matrix must not be zero. • A square matrix that has an inverse is called invertible or non-singular. • A matrix that does not have an inverse is called singular. Inverse of a second order matrix (2 x 2): a b c d A 1 1 d c det A b a Change the place of a and d and change the signs of c and b. Example: Writing simultaneous equations in matrix form Consider the simultaneous equations x + 2y = 4 3x − 5y = 1 In Matrix Form : Writing We have AX = B. This is the matrix form of the simultaneous equations. Here the unknown is the matrix X,since A and B are already known. A is called the matrix of coefficients. Solving the simultaneous equations Given AX = B, we can multiply both sides by the inverse of A, provided this exists, to give A−1AX = A−1B But A−1A = I, the identity matrix. Furthermore, IX = X, because multiplying any matrix by an identity matrix of the appropriate size leaves the matrix unaltered. So X = A−1B Example Solve the simultaneous equations x + 2y = 4 3x − 5y = 1 Solution We have already seen these equations in matrix form: