Matrices · 2.2.6
Matrix method for equations
Students write a pair of simultaneous linear equations as a matrix equation AX = B, where A holds the coefficients, X holds the unknowns, and B holds the constants. They then find A⁻¹ and calculate X = A⁻¹B, reading off the values of the unknowns directly from the resulting column matrix.
The official learning standard (2.2.6)
“Use the matrix method to solve simultaneous linear equations.”
What it means
Students write a pair of simultaneous linear equations as a matrix equation AX = B, where A holds the coefficients, X holds the unknowns, and B holds the constants. They then find A⁻¹ and calculate X = A⁻¹B, reading off the values of the unknowns directly from the resulting column matrix.
How it is examined
This is a signature Paper 2 question, typically requiring students to set up the matrix equation, find the inverse matrix, multiply to get X, and state the values of both unknowns clearly; marks are awarded at each stage, so a correct method with a small arithmetic slip still earns partial credit.
Worked example
Using the matrix method, solve the simultaneous equations x + 2y = 8 and 3x + 4y = 18.
- Write in matrix form: [[1, 2], [3, 4]] [[x], [y]] = [[8], [18]]
- Find the determinant of A: (1×4) − (2×3) = 4 − 6 = −2
- Find A⁻¹ = 1/(−2) × [[4, −2], [−3, 1]] = [[−2, 1], [1.5, −0.5]]
- Compute X = A⁻¹B: x = (−2×8) + (1×18) = −16 + 18 = 2
- y = (1.5×8) + (−0.5×18) = 12 − 9 = 3
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why use the matrix method instead of substitution or elimination?
The matrix method is a required SPM technique for this topic and is systematic, especially once you already have the inverse matrix. Substitution or elimination may still be used to check your answer, but the matrix method itself must be shown as the working method.
What if my equations are not already in the form ax + by = c?
Rearrange each equation first so that the x-term, y-term, and constant are on the correct sides, matching ax + by = c. Only after both equations are in this standard form should you write down the coefficient matrix A.
Can the matrix method solve three simultaneous equations?
The SPM syllabus for this standard focuses on two simultaneous linear equations using 2×2 matrices. The same underlying idea extends to more equations using larger matrices, but that is beyond what is required at this level.