Math-B Special Course on Matrices Completed by Mohammad Abul Hosein Contact: 01711017737 Email: ahossain69@yahoo.com

                                                      

1. Given that                           .

            Find

            (a) the possible values of x,

            (b) the value of kt.

2.         ,        .

            Find

            (a)

            (b) the determinant of A

            (c) the inverse of A.

3. Given that ,  and .

            Calculate

            (a) to one decimal place, the modulus of a.

            (b) a-c

            Given that a+kb is parallel to c,

            (c) find the value of k.

4.  The points (2,1), (5, 1) and (8, 4) are the vertices of a triangle T.

            (a) On graph paper, using a scale of 1 cm to represent 1 unit on each axis and         taking  and , draw and label the triangle T.

                        ,

            (b) Calculate the matrix product

                        .

Hence draw and label the triangle U which is the image of T under the transformation by the matrix A.

            (c) Describe fully the single transformation which is represented by the matrix A.

                        .

The triangle V is the image of U under the single transformation represented by the matrix B.

            (d) Calculate the co-ordinates of the vertices of V, and draw and label V on your                           diagram.

            (e) Describe fully the single transformation which is represented by the matrix B.

            (f) Describe fully the single transformation which is represented by the matrix                                product BA.

                       

Triangle W is the image of V under the transformation represented by the matrix C.

           

            (g) Calculate the co-ordinates of the vertices of W, and draw and label W on your                          diagram.

5. Given that  , find the value of x and y.

6.         ,         .

            (a) Find

                        (i)

                        (ii) , yhe inverse of A.

            Given that                 

            (b) find k.

7.      ,  .

            (a) Find 3A,

             Given that

           (b) Find    (i) B

                            (ii) AB

            Given that AB = kC,

            (c)  write down the value of k.   

8. A triangle U has vertices at points (1, 2), (4, 2) and (1, 3).

 (a) On graph paper using a scale of 1 unit to on each axis and taking ,       and , draw and label triangle U.

The matrix .

(b) Calculate the matrix product .

Triangle V is the image of triangle U under the transformation given by the matrix S.

(c) Draw and label V.

Triangle V is rotated through 180 about the point (0, 0) to produce triangle W.

(d) Draw and label triangle W.

(e) Determine the matrix X, representing the single transformation which maps triangle V to triangle W.

(f) Determine the matrix Y representing the single transformation which maps triangle U to triangle W.

9. Given that , Calculate .

10. (a) given that , find the value of p, q and r.

       (b) Given that , write down the matrix M.

       (c) Given that , write down the matrix N.

11.            , .

                Calculate AB.

12.  The pointa A (10,5), B  (15, 20) and C (15, 5) are the vertices of a triangle.

            a) On graph paper, using a scale of 2 cm to represent 5 units on each axis and         taking   and , draw and label .

           

            The matrix .

            b) Calculate the matrix product .

The image of  under the transformation represented bt the matrix R is ,     Where ,  and  are respecttively the images of the points A, B and C.

            c) Draw and label  on your diagram.

The transformation which matrix R represent is a rotation. Find

            d) the area, in square unite, of .

            e) the size, in degrees, of the angle of rotation.

13. Given that  and , Calculate,

            a) MN

            b) the  inverse of N.

13. The points (4, 0), (3, 2) and (4, 3) are the vertices of triangle T.

            a) Om graph paper using scale of 2 cm to represent i unit on each axis and with       and , draw and label T.

            The triangle T is transformed to another triangle U under the transformation with   matrix R where

                                                .

            b) Find the co-ordinates of the vertices of U.

            c) Draw and label U on your diagram.

            d) On your diagram, draw and label the line x+y = 4.

            The triangle T is transformed to another triangle V by means of a reflection in the line x+y = 4.

            e) On your diagram, draw and label triangle V.

            The triangle V is transformed under translation  to give triangle W.

            f) On your diagram, draw and label triangle W.

           

            g) Describe fully the single transformation which maps triangle W omnto triangle   U.