Solve the simultaneous equations $2 x + y = 7$ and $3 x - 2 y = 4$ using the matrix inverse method.

medium 2 marks Class 12 · Mathematics · Matrices

Answer

x = 2, y = 3

The correct answer is x = 2, y = 3 — Thus $x = 2$ and $y = 3$ .

Step-by-step solution

Write the coefficient matrix $A = (23 1 - 2)$ and the constant vector $B = (74)$ .
Compute $det A = 2 (- 2) - 1 \cdot 3 = - 4 - 3 = - 7 \neq = 0$ , so $A$ is invertible.
Find $A^{- 1} = \frac{1}{d e t A} (- 2 - 3 - 1 2) = \frac{1}{- 7} (- 2 - 3 - 1 2) = (\frac{2}{7} \frac{3}{7} \frac{1}{7} - \frac{2}{7})$ .
Multiply $A^{- 1} B$ : $(\frac{2}{7} \frac{3}{7} \frac{1}{7} - \frac{2}{7}) (74) = (\frac{2 \cdot 7 + 1 \cdot 4}{7} \frac{3 \cdot 7 - 2 \cdot 4}{7}) = (23)$ .
Thus $x = 2$ and $y = 3$ .

Common mistake: Multiplying the inverse on the wrong side of the equation (using

B A^{- 1}

instead of

A^{- 1} B

Concept tested: Inverse of a 2×2 matrix to solve linear equations