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0 0 The matrix ( is known as the zero matrix. 0 0) ( ) | | 9 10 The determinant of the matrix M gives the area scale factor of the associated transformation M. d –c a c 1 The inverse of M = is M–1 = –––––– , provided ad – bc ≠ 0. –b a b d ad – bc MM–1 = M–1M = I. ( ) ( ) When solving n simultaneous equations in n unknowns, the equations can be written as a matrix equation. a1 x1 x2 a M = 2 ... xn an ()() If det M ≠ 0, there is a unique solution which can be found by pre-multiplying by the inverse matrix M–1.

Where possible, find the solution set for the equations. 5 (i) (ii) Find AB, where A = ( 5 –2 k –1 3k + 8 4k + 10 3 –4 –5 and B = –2 2k + 20 3k + 25 . –2 3 4 1 –11 –14 ) ( ) Hence write down the inverse matrix A–1, stating a necessary condition on k for this inverse to exist. Using the result from part (i), or otherwise, solve the equation ( 5 –2 k 3 –4 –5 –2 3 4 x 28 y = 0 z m )( ) ( ) in each of these cases. (a) (b) (c) k = 8, giving x, y and z in terms of m k = 1 and m = 4 k = 1 and m = 2 [MEI, part, adapted] 6 Matrices A and B are given by 1 –2 0 6 6 4 A = 0 2 –2 , B = k 3 2 (where a ≠ – –32 and k ≠ 3).

17 (i) (ii) 24 (iii) Find p′ by finding the matrix product Tp. Find p′′ by finding the matrix product Sp′. Find the matrix product U = ST and show that U(p) is the same as p′′. ● How can you use the idea of successive transformations to justify the associativity of matrix multiplication: (PQ)R = P(QR)? Proving results in trigonometry Exercise 1D ● 1 Using successive transformations allows us to prove some useful results in trigonometry. If you carry out a rotation about the origin through angle α, followed by a rotation about the origin through angle β, then this is equivalent to a single rotation through the origin through angle α + β.

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A Textbook of Matrices by Hari Kishan


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