KCSE 2025 Maths P2 Q14 — Matrix Determinant & Area Scale Factor

KCSE 2025 Form 4 Matrices & Transformations

Published

The Question

“A transformation matrix T = ((a+1, 4), (4, a+1)) maps a triangle PQR of area 0.5 square units onto a triangle P'Q'R' of area 4.5 square units. Determine the possible values of a.”

1

Link area to the determinant

When a matrix transforms a shape, it multiplies the area by the size of its determinant. So the area scale factor equals the absolute value of the determinant of T. The scale factor itself is simply the new area divided by the old area.

area scale factor=∣det⁡T∣=new areaold area\text{area scale factor} = |\det T| = \frac{\text{new area}}{\text{old area}}
∣det⁡T∣=4.50.5=9|\det T| = \frac{4.5}{0.5} = 9
2

Write down the determinant of T

For a two-by-two matrix the determinant is the product of the main diagonal minus the product of the other diagonal. Both entries on the main diagonal are a+1, and both off-diagonal entries are 4, so the determinant is a+1 all squared minus 16.

det⁡T=(a+1)(a+1)−(4)(4)=(a+1)2−16\det T = (a+1)(a+1) - (4)(4) = (a+1)^{2} - 16
3

Form the equation

Because the absolute value of the determinant is 9, the determinant itself can be plus or minus 9. Taking the clean positive case, set the determinant equal to 9, then add 16 to both sides so the squared bracket stands alone.

(a+1)2−16=9(a+1)^{2} - 16 = 9
(a+1)2=25(a+1)^{2} = 25
4

Solve for a

Take the square root of both sides, remembering to keep both the positive and negative roots. This gives two values for a+1, and subtracting 1 from each gives the two possible values of a.

a+1=±5a + 1 = \pm 5
a+1=5⇒a=4a + 1 = 5 \Rightarrow a = 4
a+1=−5⇒a=−6a + 1 = -5 \Rightarrow a = -6

Final Result

The possible values are a = 4 or a = -6.

Why this method works

A matrix stretches or squashes area by a fixed factor everywhere, and that factor is exactly the absolute value of its determinant, so dividing the new area by the old area recovers it directly. Setting the determinant equal to 9 turns the geometry into an ordinary algebraic equation. The squared bracket is why two answers appear: taking a square root allows both a positive and a negative root, and each leads to a valid value of a.

✓

For a = 4, T = ((5, 4), (4, 5)) with det = 25 - 16 = 9. For a = -6, T = ((-5, 4), (4, -5)) with det = 25 - 16 = 9. Both give an area scale factor of 9, turning 0.5 into 4.5.