Vectors and matrices — Question 12

QCAA 2024, Paper 2 · 4 marks

Q12 · 2024 · Technology-activeSimple familiar4 marks

QUESTION 12 (4 marks)

A system of linear equations is given by

\begin{aligned} x-2y-2z&=-6,\\ -3x-y+z&=2,\\ 2x+3y-5z&=10. \end{aligned}

Express the system of equations as a matrix equation of the form

AX=B

, where

A

is a

3\times3

matrix and both

X

and

B

are

3\times1

column vectors.

[1 mark]

Use matrix algebra to express

X

in terms of

A

and

B

[1 mark]

Use your result from Question 12b) to determine the solution of the system of equations.

[1 mark]

Verify your result from Question 12c) using one of the given linear equations.

[1 mark]

WORKED SOLUTION

4 marks

ANSWER

(x,y,z)=(-2,3,-1)

Worked solution

\begin{bmatrix}1&-2&-2\\-3&-1&1\\2&3&-5\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} =\begin{bmatrix}-6\\2\\10\end{bmatrix}.

\mathbf X=\mathbf A^{-1}\mathbf B

c) Using a GDC,

\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}-2\\3\\-1\end{bmatrix}.

d) Substituting

x=-2

y=3

z=-1

into

x-2y-2z=-6

(-2)-2(3)-2(-1)=-6.

So the result is verified.

correctly expresses the equations in the form

AX=B

[1 mark]

correctly expresses

X

in terms of

A

and

B

[1 mark]

determines the solution to the equations

[1 mark]

verifies the result by substituting the result from 12c) into one of the given linear equations

[1 mark]

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

How many marks did you earn?

Compare your working with the guide above.