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# For what value of $\lambda$, the matrix $\begin{bmatrix} 1 & \lambda & 0 \\ 3 & -1 & 2 \\ 4 & 1 & 5 \end{bmatrix}$ is singular ?

$(A)\; \lambda = -1$
$(B)\; \lambda = 1$
$(C)\; \lambda = 0$
$(D)\; \lambda = 2$

Let$A = \begin{bmatrix} 1 & \lambda & 0 \\ 3 & -1 & 2\\ 4 & 1 & 5 \end{bmatrix}$
Since A is singular $|A| = \begin{bmatrix} 1 & \lambda & 0 \\ 3 & -1 & 2 \\ 4 & 1 & 5 \end{bmatrix}$ = 0
\begin{align*}1(-5 -2) - \lambda (15 - 8) + 0 &= 0 \\ \implies -7 -7 \lambda & = 0 \\ \implies \lambda & = -1 \end{align*}