Matrices: Solving Simultaneous Equations With the Inverse Matrix
Matrices give an IGCSE technique for solving simultaneous equations using their inverse: a pair of simultaneous equations can be written as a single matrix equation, AX = B, where A holds the coefficients, X is the column of unknowns and B is the column of constants. Multiplying both sides on the left by the inverse of A gives X = the inverse of A times B, which solves the system in one step. For a 2 by 2 matrix [a, b; c, d] the determinant is ad - bc, and the inverse is 1 divided by the determinant, multiplying the matrix [d, -b; -c, a]. The determinant carries the information about whether a solution exists: if it is zero the matrix has no inverse, the equations are either inconsistent or describe the same line, and there is no unique solution.
Before you start
Make sure you're comfortable with these topics first:
Method
- Write the equations in matrix form AX = B, putting the coefficients of x and y in order in A, the unknowns in X, and the constants in B. Line up the variables first, inserting a zero coefficient where a variable is missing.
- Calculate the determinant of A as ad - bc, and check it is not zero before going further.
- Write the inverse as 1 over the determinant, multiplying the matrix formed by swapping a and d and changing the signs of b and c.
- Multiply the inverse by B, on the LEFT, keeping the order right: X = the inverse of A times B, never B times the inverse.
- Read off x and y from the resulting column, and substitute both back into the ORIGINAL equations to check.
- If the determinant is zero, say what that means rather than continuing: the matrix is singular, so the system has either no solution or infinitely many, and the lines are parallel or identical.
Worked example
Use an inverse matrix to solve the simultaneous equations 3x + y = 11 and 4x + 2y = 16.
- Write in matrix form: A = [3, 1; 4, 2], X = the column x over y, B = the column 11 over 16.
- Find the determinant: (3)(2) - (1)(4) = 6 - 4 = 2, which is not zero, so an inverse exists.
- Write the inverse: one half times [2, -1; -4, 3], that is [1, -0.5; -2, 1.5].
- Multiply the inverse by B. Top entry: (1)(11) + (-0.5)(16) = 11 - 8 = 3.
- Bottom entry: (-2)(11) + (1.5)(16) = -22 + 24 = 2.
- So x = 3 and y = 2. Check in the original equations: 3(3) + 2 = 11 and 4(3) + 2(2) = 16, both correct.
Practice questions
Try each question, then tap to reveal the answer.
Q1Find the determinant of [5, 2; 3, 4].Show answer
Answer: (5)(4) - (2)(3) = 20 - 6 = 14.
Q2Find the inverse of [4, 1; 3, 1].Show answer
Answer: The determinant is 4 - 3 = 1, so the inverse is [1, -1; -3, 4].
Q3What does a determinant of zero tell you?Show answer
Answer: The matrix has no inverse, so the simultaneous equations have no unique solution: the lines are parallel or identical.
Q4Write the equations 2x - y = 7 and x + 3y = 0 in matrix form.Show answer
Answer: [2, -1; 1, 3] times the column x over y = the column 7 over 0.
Q5Why must the inverse be written on the left of B?Show answer
Answer: Matrix multiplication is not commutative, and only multiplying AX = B on the left by the inverse of A leaves X alone on the left-hand side.
Q6Find the inverse of [2, 0; 0, 5].Show answer
Answer: The determinant is 10, so the inverse is one tenth of [5, 0; 0, 2], which is [0.5, 0; 0, 0.2].
Q7How should a solution from an inverse matrix be checked?Show answer
Answer: Substitute the values of x and y back into both original equations and confirm each is satisfied.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
Use an inverse matrix to solve 5x + 2y = 16 and 3x + y = 9. Show your determinant and inverse clearly.
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The simultaneous equations 2x + 6y = 10 and 3x + 9y = 15 are written as a matrix equation. Show that the matrix of coefficients has no inverse, and explain what this means about the solutions.
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Free printable worksheet
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