QCE Specialist Maths: Further matrices and systems of equations Q240

Q240 · Practice questionTechnology-freeComplex unfamiliar7 marks

QUESTION 240 (7 marks)

An escape-room panel assigns nonnegative real values r, g and b to three rune types. Its calibration totals satisfy

r+g+b=12,\qquad2r+g+3b=25,\qquad r+2g+b=17.

Use Gaussian elimination on an augmented matrix to determine the three rune values.

[4 marks]

The first total is corrected to

12+\delta

; the other two totals remain unchanged. Express the values in terms of

\delta

and find the full range of

\delta

for which all are nonnegative.

[3 marks]

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WORKED SOLUTION

Practice marking scheme

7 marks

ANSWER

(a)

(r,g,b)=(1,5,6)

. (b)

(r,g,b)=(1+5\delta,5-\delta,6-3\delta)

;

-1/5\le\delta\le2

Worked solution

(a) The augmented matrix is

\left[\begin{array}{ccc|c}1&1&1&12\\2&1&3&25\\1&2&1&17\end{array}\right]

. Operations

R_2\leftarrow R_2-2R_1

and

R_3\leftarrow R_3-R_1

give rows

(1,1,1\mid12),(0,-1,1\mid1),(0,1,0\mid5)

. Swap the last rows, then add the new second row to the third, obtaining

(1,1,1\mid12),(0,1,0\mid5),(0,0,1\mid6)

. Back-substitution yields b=6, g=5 and r=1.

(b) The same elimination gives

g=5-\delta

and

b-g=1-2\delta

, so

b=6-3\delta

. The first equation then gives

r=1+5\delta

. Nonnegativity requires

\delta\ge-1/5

\delta\le5

and

\delta\le2

. Their intersection is the closed interval

[-1/5,2]

; endpoint zero values are allowed.

Equivalent mathematically justified methods accepted.

Exact answers unless a decimal accuracy is specified.

Part a: Set up the augmented matrix.

[1 mark]

Part a: Carry out valid elimination operations.

[1 mark]

Part a: Reach an echelon form.

[1 mark]

Part a: Back-substitute to obtain all three values.

[1 mark]

Part b: Find the three parameterised values.

[1 mark]

Part b: Translate all nonnegativity constraints.

[1 mark]

Part b: Intersect them and include the valid endpoints.

[1 mark]

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Further matrices and systems of equations — Question 240

QUESTION 240 (7 marks)

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