KBAT Problems
KBAT Problem: Solving with a Matrix
A KBAT problem where two real conditions become simultaneous equations, then a matrix inverse solves them, connecting a worded situation to matrix method.
Set up the equations
A shop sells two ticket types with a known total sold and total takings. That gives two equations in two unknowns, which you write as a single matrix equation.
Forming the coefficient matrix correctly is the KBAT step.
Use the inverse
Find the inverse of the 2×2 coefficient matrix (which exists because its determinant is not zero) and multiply to get the unknowns. The exam gives the inverse formula, so the marks are for setting up and substituting cleanly.
Understand the problem
A cinema sells adult and child tickets at fixed prices. On one visit, 3 adult and 2 child tickets cost RM 34; on another, 1 adult and 4 child tickets cost RM 28.
Find each ticket price. What it really asks: turn two real conditions into simultaneous equations, then solve them by the matrix inverse method rather than by elimination.
Plan and solve
- Let a be the adult price and c the child price. The conditions give 3a + 2c = 34 and a + 4c = 28.
- Write as a matrix equation: [[3, 2], [1, 4]] [a; c] = [34; 28].
- The determinant is 3×4 − 2×1 = 10, which is not zero, so the inverse exists: (1/10)[[4, −2], [−1, 3]].
- Multiply: [a; c] = (1/10)[[4, −2], [−1, 3]][34; 28] = (1/10)[136 − 56; −34 + 84] = (1/10)[80; 50] = [8; 5]. So adult = RM 8, child = RM 5.
Check and a variant
Check by substituting back: 3(8) + 2(5) = 24 + 10 = 34 and 8 + 4(5) = 8 + 20 = 28, so both conditions hold. A variant the examiner could add: use the prices to find the cost of 5 adult and 3 child tickets, which is 5(8) + 3(5) = RM 55.
Another twist is a case where the determinant is zero, meaning the two conditions clash and there is no unique solution, a good test of understanding.
Frequently asked questions
Why use the matrix method when elimination also solves it?
Because the exam question specifically asks for the matrix method and awards marks for the inverse steps. Elimination reaches the same answer but earns nothing here.
When a question names a method, use it: forming the matrix, computing the determinant and inverse, and multiplying are each worth marks you cannot get another way.
What does it mean if the determinant is zero?
A zero determinant means the matrix has no inverse, so the matrix method cannot give a unique solution. In a ticket-style problem it signals that the two conditions are really the same line or contradict each other, so there is either no solution or infinitely many.
Always compute the determinant before attempting the inverse.
Is the inverse formula given, or must I memorise it?
The inverse formula for a 2×2 matrix is on the Paper 2 formula sheet, so you do not have to memorise it, though knowing it saves time. Your marks come from applying it accurately: swap the leading diagonal, negate the other diagonal, and divide by the determinant, then multiply carefully.