Vectors and matrices — Question 16

QCAA 2022, Paper 1 · 7 marks

Q16 · 2022 · Technology-freeComplex familiar7 marks

QUESTION 16 (7 marks)

Consider this system of equations that corresponds to three planes:

x+5y=1+2z,

x+z=3y+3,

8y-\lambda=3z.

Use a Gaussian technique to determine the value of

\lambda

for which this system of equations has infinitely many solutions.

[4 marks]

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]

WORKED SOLUTION

7 marks

ANSWER

\lambda=-2

;

\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}9/4\\-1/4\\0\end{bmatrix}+k\begin{bmatrix}1/8\\3/8\\1\end{bmatrix}

k\in\mathbb R

Worked solution

a) The augmented matrix row reduction gives a final row

[0\;0\;0\mid\lambda+2]

. Hence infinitely many solutions require

\lambda=-2

b) Let

z=k

. Then

y=(3k-2)/8

and

x=k/8+9/4

. Therefore

\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}9/4\\-1/4\\0\end{bmatrix}+k\begin{bmatrix}1/8\\3/8\\1\end{bmatrix},\qquad k\in\mathbb R.

correctly rearranges the three equations

[1 mark]

establishes an augmented matrix

[1 mark]

establishes a row of zeros in the row containing

\lambda

[1 mark]

determines a value of

\lambda

[1 mark]

expresses

y

in terms of a parameter

[1 mark]

expresses

x

in terms of a parameter

[1 mark]

determines the infinite solutions expressed in the form of a vector equation of a line

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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