Edexcel GCE A Level Maths Further Maths 3 Matrices.

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1 Edexcel GCE A Level Maths Further Maths 3 Matrices. Edited by: K V Kumaran kumarmathsweebly.com

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15 0 4. A = (a) Verify that 2 is an eigenvector of A and find the corresponding eigenvalue. (b) Show that 9 is another eigenvalue of A and find the corresponding eigenvector. 2 (c) Given that the third eigenvector of A is, write down a matrix P and a diagonal matrix 2 D such that P T AP = D. [P6 June 2002 Qn 5] M 6 9 (a) Find the eigenvalues of M. A transformation T: R 2 R 2 is represented by the matrix M. There is a line through the origin for which every point on the line is mapped onto itself under T. 3. (b) Find a cartesian equation of this line. 3 A, u. 5 3 u (a) Show that det A =2(u ). (b) Find the inverse of A. a The image of the vector b when transformed by the matrix c (c) Find the values of a, b and c. 3 3 is [P6 June 2003 Qn 3] (6) [P6 June 2003 Qn 6] kumarmathsweebly.com 5

16 4. The matrix M is given by M = 3 a where p, a, b and c are constants and a > b p, c Given that MM T = ki for some constant k, find (a) the value of p, (b) the value of k, (c) the values of a, b and c, (d) det M. 5. The transformation R is represented by the matrix A, where 3 A. 3 (a) Find the eigenvectors of A. (b) Find an orthogonal matrix P and a diagonal matrix D such that (6) [P6 June 2004 Qn 5] A = PDP. (c) Hence describe the transformation R as a combination of geometrical transformations, stating clearly their order. [P6 June 2004 Qn 6] A = k (a) Show that det A = 20 4k. (b) Find A. 0 Given that k = 3 and that 2 is an eigenvector of A, (c) find the corresponding eigenvalue. (6) kumarmathsweebly.com 6

17 Given that the only other distinct eigenvalue of A is 8, (d) find a corresponding eigenvector. [FP3/P6 June 2005 Qn 7] 7. A transformation T : R 2 R 2 is represented by the matrix A = 2 2, where k is a constant. 2 Find (a) the two eigenvalues of A, (b) a cartesian equation for each of the two lines passing through the origin which are invariant under T. 8. A = k 0 9 [*FP3/P6 January 2006 Qn 3] 2 k, where k is a real constant. 0 (a) Find values of k for which A is singular. Given that A is non-singular, (b) find, in terms of k, A. [FP3/P6 January 2006 Qn 4] 9. A = Prove by induction, that for all positive integers n, A n = 0 0 n 0 2 ( n 3n) n. 2 [FP3 June 2006 Qn ] kumarmathsweebly.com 7

18 0. The eigenvalues of the matrix M, where are and 2, where < 2. M = 4 2, (a) Find the value of and the value of 2. (b) Find M. (c) Verify that the eigenvalues of M are and 2. A transformation T : R 2 R 2 is represented by the matrix M. There are two lines, passing through the origin, each of which is mapped onto itself under the transformation T. (d) Find cartesian equations for each of these lines. 0. Given that is an eigenvector of the matrix A, where 3 4 p A q (a) find the eigenvalue of A corresponding to, (b) find the value of p and the value of q. [FP3 June 2006 Qn 5] The image of the vector l m n when transformed by A is (c) Using the values of p and q from part (b), find the values of the constants l, m and n. [FP3 June 2007 Qn 3] kumarmathsweebly.com 8

19 2. where p and q are constants. M = 0 2 p 3 p 2 q, Given that 2 (a) show that q = 4p. is an eigenvector of M, Given also that λ = 5 is an eigenvalue of M, and p < 0 and q < 0, find (b) the values of p and q, (c) an eigenvector corresponding to the eigenvalue λ = M = [FP3 June 2008 Qn 2] (a) Show that 7 is an eigenvalue of the matrix M and find the other two eigenvalues of M. (b) Find an eigenvector corresponding to the eigenvalue M = 0 2 k 0 6 Given that is an eigenvector of M, 6 (a) find the eigenvalue of M corresponding to (b) show that k = 3, (c) show that M has exactly two eigenvalues. 3, where k is a constant. 6, 6 A transformation T : R 3 R 3 is represented by M. [FP3 June 2009 Qn 3] kumarmathsweebly.com 9

20 The transformation T maps the line l, with cartesian equations line l2. x 2 = y 3 = z, onto the 4 (d) Taking k = 3, find cartesian equations of l2. [FP3 June 200 Qn 6] 5. The matrix M is given by k M = 0, k. 3 2 (a) Show that det M = 2 2k. (b) Find M, in terms of k. The straight line l is mapped onto the straight line l 2 by the transformation represented by the matrix The equation of l 2 is (r a) b = 0, where a = 4i + j+ 7k and b = 4i + j+ 3k. (c) Find a vector equation for the line l. 6. The matrix M is given by 2 0 M = (a) Show that 4 is an eigenvalue of M, and find the other two eigenvalues. (b) For the eigenvalue 4, find a corresponding eigenvector. [FP3 June 20 Qn 7] The straight line l is mapped onto the straight line l2 by the transformation represented by the matrix M. The equation of l is (r a) b = 0, where a = 3i + 2j 2k and b = i j+ 2k. (c) Find a vector equation for the line l2. [FP3 June 202 Qn 8] kumarmathsweebly.com 20

21 7. It is given that 2 0 is an eigenvector of the matrix A, where A 2 b 0 a 8 and a and b are constants. (a) Find the eigenvalue of A corresponding to the eigenvector 2. 0 (b) Find the values of a and b. (c) Find the other eigenvalues of A. 8. The matrix M is given by [FP3 June 203_R Qn 6] a M 2 b c, where a, b and c are constants. 0 (a) Given that j + k and i k are two of the eigenvectors of M, find (i) the values of a, b and c, (ii) the eigenvalues which correspond to the two given eigenvectors. (8) (b) The matrix P is given by 0 P 2 d, where d is constant, d 0 Find (i) the determinant of P in terms of d, (ii) the matrix P in terms of d. [FP3 June 203 Qn 5] kumarmathsweebly.com 2

22 9. The symmetric matrix M has eigenvectors 2 2, 2 2 and 2 2 with eigenvalues 5, 2 and respectively. 20. (a) Find an orthogonal matrix P and a diagonal matrix D such that P T MP = D Given that P = P T (b) show that M = PDP (c) Hence find the matrix M. 0 2 M [FP3 June 204_R Qn 6] (a) Show that matrix M is not orthogonal. (b) Using algebra, show that is an eigenvalue of M and find the other two eigenvalues of M. (c) Find an eigenvector of M which corresponds to the eigenvalue. The transformation M : 3 3 is represented by the matrix M. (d) Find a cartesian equation of the image, under this transformation, of the line y z x 2 [FP3 June 204 Qn 2] kumarmathsweebly.com 22

23 2. A = (a) Find the eigenvalues of A. (b) Find a normalised eigenvector for each of the eigenvalues of A. (c) Write down a matrix P and a diagonal matrix D such that P T AP = D. 2 3 A = k 3, where k is a constant 2 k [FP3 June 205 Qn 3] Given that the matrix A is singular, find the possible values of k. [FP3 June 206 Qn ] p M = 2 6 2, 0 2 q where p and q are constants. 2 Given that 2 is an eigenvector of the matrix M, (a) find the eigenvalue corresponding to this eigenvector, (b) find the value of p and the value of q. Given that 6 is another eigenvalue of M, (c) find a corresponding eigenvector. Given that 2 is a third eigenvector of M with eigenvalue 3, 2 (d) find a matrix P and a diagonal matrix D such that P T MP = D. [FP3 June 206 Qn 6] kumarmathsweebly.com 23

24 24. The matrix M is given by æ M = ç ç è (a) Show that det M = 2k. (b) Find M in terms of k. k ö, k R, k 2 ø The straight line l is mapped onto the straight line l2 by the transformation represented by the matrix æ 0 0 ö ç 2-2 ç è -4 - ø Given that l2 has cartesian equation x - 5 = y = z - 3 (c) find a cartesian equation of the line l 25. A non-singular matrix M is given by 3 k 0 M k 2 0, where k is a constant. k 0 (a) Find, in terms of k, the inverse of the matrix M. (6) [FP3 June 207 Qn 6] The point A is mapped onto the point ( 5, 0, 7) by the transformation represented by the matrix (b) Find the coordinates of the point A. [F3 IAL June 204 Qn 4] kumarmathsweebly.com 24

25 26. Given that M 4 k, where k is a constant. 0 3 is an eigenvector of the matrix M, (a) find the eigenvalue of M corresponding to 7 9, (b) show that k = 7, (c) find the other two eignevalues of the matrix M. The image of the vector p q r under the transformation represented by M is (d) Find the values of the constants p, q and r. 27. k 0 M=, where k is a constant k 3 (a) Find M in terms of k. Hence, given that k = 0 (b) find the matrix N such that MN = [F3 IAL June 205 Qn 4] [F3 IAL June 206 Qn 4] kumarmathsweebly.com 25

26 28. æ A = ç ç è - 3 a ö æ 2 0 4, B = ç ç ø è 2 b ö ø where a and b are constants. (a) Write down A T in terms of a. (b) Calculate AB, giving your answer in terms of a and b. (c) Hence show that (AB) T = B T A T () [F3 IAL June 207 Qn 2] 29. æ M = ç ç è ö ø (a) Show that 6 is an eigenvalue of the matrix M and find the other two eigenvalues of M. (b) Find a normalised eigenvector corresponding to the eigenvalue 6. [F3 IAL June 207 Qn 4] kumarmathsweebly.com 26