Q.Find $\lambda$, if the matrix $\begin{pmatrix} 1 & 1 & 3 \\ 2 & \lambda & 4 \\ 9 & 7 & 11 \end{pmatrix}$ has no inverse.

The inverse fails to exist iff the matrix is singular ($\det = 0$). Expanding the determinant gives $-16\lambda + 28 = 0$, so $\lambda = \dfrac{7}{4}$.

Step 1 — Condition for no inverse. A matrix $A$ has no inverse $\iff \det A = 0$.

Step 2 — Expand the determinant along the first row.

$\det = 1\,(\lambda\cdot 11 - 4\cdot 7) - 1\,(2\cdot 11 - 4\cdot 9) + 3\,(2\cdot 7 - \lambda\cdot 9).$

$= (11\lambda - 28) - (22 - 36) + 3(14 - 9\lambda).$ …