EXM.1.AHL.TZ0.45

pestleMathematicsAIHLPaper 1EXM· ahl-1-14-introduction-to-matricessource ↗

Let A = $\left( {\begin{array}{*{20}{c}} 1&2&3 \\ 2&{ - 1}&2 \\ 3&{ - 3}&2 \end{array}} \right)$ , D = $\left( {\begin{array}{*{20}{c}} { - 4}&{13}&{ - 7} \\ { - 2}&7&{ - 4} \\ 3&{ - 9}&5 \end{array}} \right)$ , and C = $\left( {\begin{array}{*{20}{c}} 5 \\ 7 \\ {10} \end{array}} \right)$ .

Given matrices A, B, C for which AB = C and det A ≠ 0, express B in terms of A and C.

[2]

Find the matrix DA.

[1]

b.i.

Find B if AB = C.

[2]

b.ii.

Find the coordinates of the point of intersection of the planes $x + 2y + 3z = 5$ , $2x - y + 2z = 7$ , $3x - 3y + 2z = 10$ .

[2]

Markscheme / solution

Since det A ≠ 0, A^–1 exists. (M1)

Hence AB = C ⇒ B = A^–1C (C1)

[2 marks]

DA = $\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right)$ (A1)

[1 mark]

b.i.

B = A^–1C = DC (M1)

$= \left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \\ 2 \end{array}} \right)$ (A1)

[2 marks]

b.ii.

The system of equations is $\begin{array}{*{20}{c}} {x + 2y + 3z = 5} \\ {2x - y + 2z = 7} \\ {3x - 3y + 2z = 10} \end{array}$

or A $\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) =$ C (M1)

The required point = (1, –1, 2). (A1)

[2 marks]

Examiners’ report

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b.i.

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b.ii.

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