Matrices: Exam-Pattern Drill (6)

Question

If $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$, find $A^{-1}$.

Solution — Step by Step

Final answer: $A^{-1} = \tfrac{1}{5}\begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix}$.

Why This Works

For $2\times 2$ matrices, “swap the diagonal, negate the off-diagonal, divide by determinant” is a memorable mantra. CBSE Class 12 boards almost always test this exact pattern in 2-mark or 4-mark questions.

For larger matrices (3×3, 4×4), use the adjugate-divided-by-determinant or Gauss-Jordan elimination — but JEE Main rarely goes beyond $2 \times 2$ inversion under exam pressure.

Alternative Method

Use row reduction: $[A \mid I] \to [I \mid A^{-1}]$. Slower for $2\times 2$ but essential for $3\times 3$ matrices in CBSE 5-mark questions.

Common Mistake