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(D) matrix $A$ has no inverse if and only if its determinant is zero,i.e.,$|A| = 0$. Calculating the determinant of $A$: $|A| = \begin{vmatrix} \lambd

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infinite

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(D) matrix $A$ has no inverse if and only if its determinant is zero,i.e.,$|A| = 0$. Calculating the determinant of $A$: $|A| = \begin{vmatrix} \lambda - 1 & \lambda & \lambda + 1 \ 2 & -1 & 3 \ \lambda + 3 & \lambda - 2 & \lambda + 7 \end{vmatrix}$ Apply the row operation $R_3 ightarrow R_3 - R_1$: $|A| = \begin{vmatrix} \lambda - 1 & \lambda & \lambda + 1 \ 2 & -1 & 3 \ 4 & -2 & 6 \end{vmatrix}$ Notice that row $3$ is exactly $2$ times row $2$ $(R_3 = 2R_2)$. Since two rows are proportional,the determinant of the matrix is $0$ for all real values of $\lambda$. Therefore,the matrix $A$ is singular for all $\lambda \in \mathbb{R}$. Thus,there are infinitely many values of $\lambda$ for which the matrix has no inverse.

Number of real values of $\lambda$ for which the matrix $A = \begin{bmatrix} \lambda - 1 & \lambda & \lambda + 1 \\ 2 & -1 & 3 \\ \lambda + 3 & \lambda - 2 & \lambda + 7 \end{bmatrix}$ has no inverse.

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