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medium difficulty9 min

Solving Linear Equations Using Matrix Inverse

A system of linear equations can be written as AX = B. If |A| ≠ 0, the unique solution is X = A−1B.

In one line: Coefficients times variables equals constants: AX = B.

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Understand it

  • In the matrix method, the coefficient matrix A must match the fixed order of variables in X and the equation order used in B.
  • Once the system is written as AX = B, the inverse method is valid only when A is square and |A| ≠ 0.
  • Multiplying both sides on the left by A−1 gives A−1AX = A−1B, so IX = A−1B and therefore X = A−1B.
  • This left multiplication is important because matrix multiplication is not generally commutative; B A−1 is either undefined or incorrect in most exam setups.
  • If |A| = 0, the system must be tested for consistency instead of using the inverse formula for a unique answer.

Example

The equations 2x + y = 5 and x - y = 1 give A = [[2,1],[1,-1]], X = [[x],[y]], B = [[5],[1]]. Since |A| = -3 ≠ 0, X = A−1B gives x = 2 and y = 1.

Formula and rule

Write the system as AX = B.

If |A| ≠ 0, then A−1 exists and X = A−1B.

If |A| = 0, do not apply the inverse formula for a unique solution.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Solve 2x + y = 5 and x - y = 1.

  1. 1A = [[2,1],[1,-1]], B = [[5],[1]]. |A| = 2(-1) - 1(1) = -3.
  2. 2A−1 = 1/(-3)[[-1,-1],[-1,2]] = [[1/3,1/3],[1/3,-2/3]].
  3. 3X = A−1B = [[1/3,1/3],[1/3,-2/3]][[5],[1]] = [[2],[1]].
  4. 4Hence x = 2, y = 1.

Method you can reuse

  1. 1Arrange equations in a fixed variable order.
  2. 2Form A, X, and B.
  3. 3Compute |A|.
  4. 4If non-zero, find A−1.
  5. 5Multiply A−1B in the correct order.
  6. 6State the values of variables and, if time permits, substitute back to check.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

For AX = B, what is X if |A| ≠ 0?

Think first, then show answer
If |A| ≠ 0, A inverse exists, so multiply AX = B on the left by A−1 to get X = A−1B.

Common wrong answer

Multiplying BA−1 or changing the order of variables midway.

Don't mix this up

Writing the constant matrix B incorrectly by changing the order of equations or constants.

Exam tip

Always define A, X, and B clearly before applying X = A−1B; this earns method credit and reduces arrangement errors.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may ask students to solve two or three simultaneous linear equations using the matrix method, or to use a real-life case converted into linear equations.
  1. 1

    Enough for 1 mark

    A system of linear equations can be written as AX = B. If |A| ≠ 0, the unique solution is X = A−1B.

  2. 2

    Add for 2 marks

    Example: The equations 2x + y = 5 and x - y = 1 give A = [[2,1],[1,-1]], X = [[x],[y]], B = [[5],[1]]. Since |A| = -3 ≠ 0, X = A−1B gives x = 2 and y = 1.

  3. 3

    Add for 3 marks

    In the matrix method, the coefficient matrix A must match the fixed order of variables in X and the equation order used in B.

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