Q.Using matrix method solve the following system of equations: $2x + 5y = 1$ $3x + 2y = 7$

By the matrix (inversion) method, $x = 3$ and $y = -1$.

Write the system $2x + 5y = 1,\; 3x + 2y = 7$ in matrix form $AX = B$:

$A = \begin{pmatrix} 2 & 5 \\ 3 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 7 \end{pmatrix}.$

Step 1 — Determinant of $A$:

$|A| = (2)(2) - (5)(3) = 4 - 15 = -11 \neq 0,$

so $A$ is non-singular and $A^{-1}$ exists.

Step 2 — Inverse of $A$. For a $2\times2$ matrix, $A^{-1} = \dfrac{1}{|A|}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$:

$A^{-1} = \frac{1}{-11}\begin{pmatrix} 2 & -5 \\ -3 & 2 \end{pmatrix}.$

Step 3 — $X = A^{-1}B$: …