Simultaneous equations by matrices — A/L Combined Mathematics

What must I remember about Simultaneous equations by matrices?

• Matrix formulation of linear system: System a1*x + b1*y = d1 and a2*x + b2*y = d2 is written in matrix form as A * X = B, where coefficient matrix A = [a1 b1; a2 b2], variable vector X = [x; y], and constant vector B = [d1; d2].
• Matrix Inversion method: If det(A) ≠ 0, multiply both sides on the left by A^-1: A^-1(AX) = A^-1B => I*X = A^-1B => X = A^-1B.
• Solution computation: [x; y] = (1/(a1*b2 - a2*b1)) * [b2 -b1; -a2 a1] * [d1; d2].
• Cramer's Rule for 2×2 system: x = Δx / Δ and y = Δy / Δ, where Δ = det(A), Δx = det([d1 b1; d2 b2]), and Δy = det([a1 d1; a2 d2]), provided Δ ≠ 0.
• Nature of solutions: Unique solution if det(A) ≠ 0; if det(A) = 0 and Δx = Δy = 0, system has Infinitely Many Solutions (consistent, dependent equations representing coincident lines); if det(A) = 0 and at least one of Δx, Δy ≠ 0, system has No Solution (inconsistent, parallel distinct lines).

Test yourself (1 minute)

1. Pre-multiplication requirement

   In solving AX = B, you must PRE-MULTIPLY by A^-1: A^-1(AX) = A^-1B; post-multiplying B*A^-1 is undefined because dimensions do not match.

2. Homogeneous linear systems

   AX = O always has the trivial solution X = O (x = 0, y = 0); non-trivial (non-zero) solutions exist if and only if det(A) = 0.

3. Consistency test

   A linear system is Consistent if it possesses at least one solution; Inconsistent if no solution exists.

Practise Simultaneous equations by matrices questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 24

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Year 2 · Term 1