Solved: The matrices A and B are defined by A = ( ... [IB Mathematics Applications & Interpretation (Math AI)]

Question

HLPaper 2

The matrices A and B are defined by $A = \left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 2&4 \end{array}} \right)$ and $B = \left( {\begin{array}{*{20}{c}} { - 1}&0 \\ 0&1 \end{array}} \right)$ .

Triangle X is mapped onto triangle Y by the transformation represented by AB. The coordinates of triangle Y are (0, 0), (−30, −20) and (−16, 32).

Describe fully the geometrical transformation represented by B.

[2]

Verified

Solution

reflection in the y-axis A1A1

[2 marks]

Find the coordinates of triangle X.

[5]

Verified

Solution

$X = {\left( {AB} \right)^{ - 1}}Y$ M1

EITHER

$AB = \left( {\begin{array}{*{20}{c}} { - 3}&{ - 2} \\ { - 2}&4 \end{array}} \right)$ , so ${\left( {AB} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} { - \frac{1}{4}}&{ - \frac{1}{8}} \\ { - \frac{1}{8}}&{\frac{3}{{16}}} \end{array}} \right)$ M1A1

$X = {B^{ - 1}}{A^{ - 1}}Y$ M1A1

THEN

$X = \left( {\begin{array}{*{20}{c}} 0&{10}&0 \\ 0&0&8 \end{array}} \right)$ (A1)

So the coordinates are (0, 0), (10, 0) and (0, 8). A1

[5 marks]

Find the area of triangle X.

[2]

Verified

Solution

$\frac{{10 \times 8}}{2} = 40$ units² M1A1

[2 marks]

Hence find the area of triangle Y.

[3]

Verified

Solution

$\det \left( {AB} \right) = - 16$ M1A1

Area $= 40 \times 16 = 640$ units² A1

[3 marks]

Matrix A represents a combination of transformations:

A stretch, with scale factor 3 and y-axis invariant;
Followed by a stretch, with scale factor 4 and x-axis invariant;
Followed by a transformation represented by matrix C.

Find matrix C.

[4]

Verified

Solution

A stretch, with scale factor 3 and y-axis invariant is given by $\left( {\begin{array}{*{20}{c}} 3&0 \\ 0&1 \end{array}} \right)$ A1

A stretch, with scale factor 4 and x-axis invariant is given by $\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&4 \end{array}} \right)$ A1

So $C = A{\left( {\begin{array}{*{20}{c}} 3&0 \\ 0&1 \end{array}} \right)^{ - 1}}{\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&4 \end{array}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} 1&{ - \frac{1}{2}} \\ {\frac{2}{3}}&1 \end{array}} \right)$ M1A1

[4marks]

The matrices A and B are defined by A = ( 3 − 2 2 4 ) and B = ( − 1 0 0 1 ) .Triangle X is mapped onto triangle Y by the transformation represented by AB. The coordinates of triangle Y are (0, 0), (−30, −20) and (−16, 32).

Related topics