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Matrices

1.       Given the matrix A =  find the values of the real number k for which det(AkI) = 0
where I = .

Working:

 

 

Answer:

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(Total 4 marks)

 


 

2.       (a)     Find the values of a and b given that the matrix A =  is the inverse of the matrix B =

(b)     For the values of a and b found in part (a), solve the system of linear equations

          x + 2y – 2z = 5
3x + by + z = 0
x + y – 3z = a – 1.

Working:

 

 

Answers:

(a)       ..................................................................

(b)       ..................................................................

(Total 4 marks)

 


 

3.       Find the values of the real number k for which the determinant of the matrix  is equal to zero.

Working:

 

 

Answer:

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(Total 3 marks)

 

 

4.       (a)     Given matrices A, B, C for which AB = C and det A ¹ 0, express B in terms of A and C.

(2)

(b)     Let A = , D =  and C = .

(i)      Find the matrix DA;

(ii)     Find B if AB = C.

(3)

(c)     Find the coordinates of the point of intersection of the planes
x + 2y + 3z = 5, 2x y + 2z = 7 and 3x – 3y + 2z = 10.

(2)

(Total 7 marks)

 


 

5.       If A =and B = , find 2 values of x and y, given that AB = BA.

Working:

 

 

Answer:

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(Total 3 marks)

 

 

6.       The matrix  is singular. Find the values of k.

Working:

 

 

Answer:

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(Total 3 marks)

 


 

7.       The matrix A is given by

          A =

          Find the values of k for which A is singular.

Working:

 

 

Answer:

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(Total 6 marks)

 

 

8.       (a)     Find the determinant of the matrix

(1)

(b)     Find the value of λ for which the following system of equations can be solved.

(3)

(c)     For this value of λ, find the general solution to the system of equations.

(3)

(Total 7 marks)

 


 

9.       Given that A =  and I = , find the values of λ for which (A λI) is a singular matrix.

Working:

 

 

Answer:

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(Total 6 marks)

 

 

10.     The matrices A, B, C and X are all non-singular 3 × 3 matrices.
Given that AlXB = C, express X in terms of the other matrices.

Working:

 

 

Answer:

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(Total 6 marks)

 

 

11.     Given that the matrix A =  is singular, find the value of p.

Working:

 

 

Answer:

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(Total 6 marks)

 

 

12.     The square matrix X is such that X3 = 0. Show that the inverse of the matrix (I X) is
I + X + X2.

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(Total 6 marks)

 

 

13.     Given that A =  and B =  find X if BX = A AB.

(Total 6 marks)

 

 

14.     (a)     Let M = . Find det M.

(2)

(b)     Find the values of k for which the following system of equations does not have a unique solution.

                          xky + 3z = –1

                          4x + 5y + z = 2

                            xy + kz = 1

(3)

(Total 5 marks)

 

 

15.     Let A=  and B = , where h and k are integers. Given that det A = det B and that det AB = 256h,

(a)     show that h satisfies the equation 49h2 – 130h + 81 = 0;


 

(b)     hence find the value of k.

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(Total 6 marks)

 


 

16.     Consider the system of equations T , where T = .

(a)     Find the solution of the system when r = 0 and s = 3.

(4)

(b)     The solution of the system is not unique.

(i)      Show that s =  r2.

(ii)     When r = 2 and s = 18, show that the system can be solved, and find the general solution.

(11)

(c)     Use mathematical induction to prove that, when r = 0,

          Tn = , nÎ +.

(9)

(Total 24 marks)

 


 

17.     (a)     Find the inverse of the matrix .

(b)     Hence solve the system of equations

                                 x + 2y + z = 0

                                x + y + 2z = 7

                               2x + y + z = 17

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(Total 6 marks)

 


 

18.     Let A =  and X =  Given that AX = kX, where kÎ , find the values of k for which there is an infinity of solutions for X.

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(Total 6 marks)

 


 

19.     Consider the system of equations A  =  where A =  and kÎ .

(a)     Find det A.

(b)     Find the set of values of k for which the system has a unique solution.

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(Total 6 marks)

 


 

20.     Let A =

(a)     Find the values of l for which the matrix (AlI) is singular.

(5)

Let A2 + mA+ nI = O where m, nÎ  and O =

(b)     (i)      Find the value of m and of n.

(ii)     Hence show that I =  A (6IA).

(iii)    Use the result from part (b) (ii) to explain why A is non-singular.

(12)

(c)     Use the values from part (b) (i) to express A4 in the form pA+ qI where p, qÎ .

(5)

(Total 22 marks)

 


 

21.     Determine the values of k for which  is singular.

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(Total 6 marks)

 


 

22.     Let M be the matrix

Find all the values of a for which M is singular.

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(Total 6 marks)

 


 

23.     Let M2 = M where M =

(a)     (i)      Show that a + d = 1.

(ii)     Find an expression for bc in terms of a.

(5)

(b)     Hence show that M is a singular matrix.

(3)

(c)     If all of the elements of M are positive, find the range of possible values for a.

(3)

(d)     Show that (IM)2 = I M where I is the identity matrix.

(3)

(e)     Prove by mathematical induction that (I − M)n = I − M for nÎ+.

(6)

(Total 20 marks)

 

 

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