Matrices & Transformations (STEP 2/3)

Topic overview

STEP 2: 2×2 matrices, determinants, inverse matrices, transformations and invariant lines; STEP 3 additionally includes 3×3. Eigenvalues/diagonalisation are not required without provided guidance.

Key concepts

Verify STEP 2/3 separately. Complex numbers and 2×2 matrices may belong to STEP 2; De Moivre, general complex roots, planes and vector products, etc. belong to STEP 3.

Key points

◆Follow the prerequisite knowledge requirements given by the examination body; new concepts may only be used when the question provides sufficient guidance.
◆The examples on this site are training with a limited scope and do not represent full syllabus coverage or official difficulty.

Worked examples

Try each example yourself before checking the explanation.

1Written examples within the scope

Let $I_n = \displaystyle\int_0^{\pi/2} \sin^n x\, dx$ for non-negative integer $n$ .

Show that $I_0 = \dfrac{\pi}{2}$ and $I_1 = 1$ .

Show that $I_n = \dfrac{n-1}{n}\,I_{n-2}$ for $n \ge 2$ (reduction formula).

Show that $I_{2m} = \dfrac{(2m-1)!!}{(2m)!!}\cdot\dfrac{\pi}{2}$ , where $(2m)!! = 2\cdot4\cdot6\cdots(2m)$ .

Hence evaluate $I_6$ and $I_7$ .

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