Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — finding an inverse and checking it
Find the inverse of A=(3512).
- 1detA=(3)(2)−(1)(5)=6−5=1Leading diagonal minus the other diagonal. Check this first — a zero would mean no inverse.
- 2Swap the leading diagonal and change the signs of the other two.a and d swap; b and c only change sign.
- 3A−1=11(2−5−13)
- 4A−1=(2−5−13)
- 5Check: AA−1=(1001) ✓The identity matrix. If you do not get it, the inverse is wrong.
Example 2 — simultaneous equations by matrix
Use a matrix method to solve 3x+y=11 and 5x+2y=18.
- 1Write in the form Ax=b.Coefficients on the left, constants on the right.
- 2(3512)(xy)=(1118)
- 3A−1=(2−5−13), since detA=1.
- 4(xy)=(2−5−13)(1118)The inverse goes on the left of both sides.
- 5=(22−18−55+54)=(4−1)
- 6x=4, y=−1Check: 3(4)−1=11 ✓ and 5(4)−2=18 ✓
Example 3 — proving three points are collinear
Show that A(1,2), B(3,5) and C(7,11) are collinear.
- 1AB=(35)−(12)=(23)Destination minus start.
- 2BC=(711)−(35)=(46)
- 3(46)=2(23), so BC=2ABOne is a scalar multiple of the other, so they are parallel.
- 4Both vectors pass through B.Parallel alone is not enough — the shared point is essential.
- 5Therefore A, B and C lie on the same straight line.State the conclusion explicitly; it carries a mark.