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QUESTION 6 Solve the matrix equation for X. 3% 27l 1 o |00 @ 1 4 o [3 ® 1 5 13 —14 2 ln 12J 54 56 (D) l }
QUESTION 5 The augmented matrix shown is produced when a Gaussian elimination technique is used to solve a certain system of equations with three variables. 1 3|4 0 -1 5 |[-6 0 0 110 The geometric interpretation of the solution to this system of equations is best represented by (A) % | é ©) J J (D) E i
QUESTION 16 (6 marks) 1 2 2 LetA=|2 1 1 2 2 1 Given A* = pA® +gA? +rA + 31 use matrix algebra to determine the value of the scalars p ¢ and r. p q g P-4
QUESTION 16 (7 marks) Consider this system of equations that corresponds to three planes. x+5y=14+2z x+z=3y+3 8y—A=3z a) Use a Gaussian technique to determine the value of 4 for which this system of equations has infinitely many solutions. [4 marks] b) Use the result from Question 16a) to determine the infinitely many solutions. Express your answer in the form of a vector equation of a line. [3 marks]
QUESTION 9 Consider the matrix equation. Matrix X 1s T 1 ] — AN 402 — N wn wvw wn n S o~ N N <t n N A | ~~ ~ ~ ~ < /m O a ~ —~ ~ N’
QUESTION 13 (7 marks) Consider the following system of equations. x+2y+3z=4 3x —2y+3z=5 2x +4y+6z =k a) Use a Gaussian technique to determine the value of k for which there are infinitely many solutions. [3 marks] Consider the original system of equations when k = 3. b) Determine the number of solutions of this system of equations. Justify your conclusion. [2 marks] c) Use your result from 13b) to describe a geometric interpretation of this system of equations. [2 marks]
QUESTION 20 (7 marks) A cubic function of the form y = ax3® + bx? + cx passes through the points (2 2) (3 1) and (5 5). a) Use matrix algebra to determine the values of a b and c. [5 marks] b) Evaluate the reasonableness of your result from 20a). [2 marks]
QUESTION 9 The geometric interpretation of a certain system of three equations with no solution is shown. Given two of the equations are x + y—z=0.5 and x —y —z=0.5 the third equation could be (A) 2x-2y-2z=1 (B) 2x+2y-—2z=1 (C) 2x-2y+2z=3 (D) 2x+2y-—2z=3
QUESTION 12 (5 marks) 1 -2 0 2 -1 -l Given A= f >} B= t ‘ and C -( 0 3} determine X in the matrix equation XA—-XC=B.
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