Practical - Practice questions - Matrices PDF

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PRACTICE QUESTIONS – MATRICES...

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1 202BP S

PRACTICE QUESTIONS 1. If the matrices A and B are given by

–

!2 3 $ A = # " 4 1&%

MATRICES !3 -1$ B = # " 2 0 &%

(a) Calculate 3A - 2B + 7I (b) 3A + 4X = B is an equation for an unknown matrix X. Find the matrix X. 2 2 (c) Calculate the matrix products AB, BA, AA ( = A ) and BB ( = B ).

2.

Suppose that J and K are defined as " 0 2 3% K = $ -1 2 -5' ' $ $# 0 1 4 '&

" 1 3 7% J = $ # !2 4 1 '&

2 2 (a) Which of the four products J K, KJ, J and K make sense ? If they make sense, calculate these products. T T (b) Find J and K .

3.

Suppose that P is a 3 ! 2 matrix, Q is a 2 ! 2 matrix and that R is a 3 ! 3 matrix. Which of the following matrix products make sense ? If they make sense, what are the dimensions of the product matrix ? (i)

P2

(ii) PQ

(iii) PR

(iv) QP

(v)

Q2

(vi) QR

(vii) RP

(viii)R Q

(x) PQR

(xi) RPQ

(xii) QRP.

(ix) R2

4.

Draw three dimensional diagrams that represent the following transformations (a) a reflection in the xz plane (b) a reflection in the yz plane. From these diagrams find where the point with co-ordinates (x,y,z) ends up. Use this to find the 3 ! 3 matrices that represent each of these transformations. Use matrix products to find the matrix that represents the combined transformation - a reflection in the xz plane followed by a reflection in the yz plane.

5.

Find the matrix that represents the following sequence of transformations in two dimensions: First: Rotate all points by an angle -a Next: Reflect all points across the x-axis Last: Rotate all points clockwise by an angle a.

6.

What is the effect of the matrix A on the unit square when A is given by (a)

7.

! 3 -1$ A = # " 0 1 &%

(b )

3x + 2y + z = 2 4x + 2y + 2z = 8 x - y + z = 4

2x + 4y + 6z = -12 2x - 3y - 4z = 15 3x + 4y + 5z = -8.

Find the determinants and inverses, if they exist, of each of the following matrices : ! 1 2 -3$ (b ) # 1 -2 1 & & # #" 5 -2 -3&%

!1 2 1$ (a) #1 3 2& & # #"1 0 1&%

9.

(b)

Solve each of the following sets of equations by Gaussian elimination: (a)

8.

! 2 3$ A = # & " 4 1%

(c)

! 2 1 0$ # 0 1 2& . & # #" 2 5 6&%

Use your answer to Q8(c) to solve: 2x +

y

= 4

y + 2z = 0 2x + 5y + 6z = 6.

10. Find the eigenvalues and eigenvectors of each of the following matrices " 1 1% (a) $ ' # !2 4&

"0 2 % (b) $ ' # 2 !3&

(c)

:

" 3 0 2% $0 1 0' . ' $ #$2 0 3'&

11. Without solving the following sets of equations, determine if either of these sets have nonzero solutions. Where the solutions are nonzero, find these solutions using Gaussian elimination. (a)

12.

2x - y + 3z = 0 3x + 2y + z = 0 x - 4y + 5z = 0

(b )

x + 2y + 3z = 0 2x + y + 3z = 0 3x + 2y + z = 0.

Using Gaussian elimination, balance the following chemical reactions: (a)

Pb (N 3 )2 + Cr(MnO4 ) 2 ! Cr2 O 3 + MnO2 + Pb3O 4 + N2

(b )

As + KNO 3 + HNO3 ! KAsO4 + NO + H2 O ....