Inverse Of A Matrix

The inverse of a two by two matrix is the adjoint divided by the determinant, a standard JEE Advanced computation valid only when the determinant is non-zero. For A with rows (4,7),(2,6), the determinant is 4·6 - 7·2 = 24 - 14 = 10, which is non-zero so the inverse exists. The adjoint swaps the diagonal entries and negates the off-diagonal entries, giving adj(A) with rows (6, -7),(-2, 4). Dividing by the determinant 10, the inverse has rows (6/10, -7/10),(-2/10, 4/10). The top-left entry is 6/10 = 3/5 in simplest form. Option 2/5 wrongly takes the bottom-right entry 4/10 without swapping diagonals. Option -7/10 is the top-right off-diagonal entry, not the top-left. Option 1/4 ignores the determinant scaling and merely reciprocates the (1,1) entry of A. Hence the top-left entry is 3/5. Plausibility check: multiplying A by the proposed inverse should yield the identity, and the (1,1) entry 4·(6/10) + 7·(-2/10) = (24 - 14)/10 = 1 confirms the inverse is correct.