Number of real values of λ 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
Step-by-Step Solution
Key Concept: A matrix has no inverse if its determinant is equal to zero. Calculate the determinant of the given 3x3 matrix and set it to zero to find the values of \lambda.
A matrix A has no inverse if |A| = 0. <br> |A| = (\lambda-1)[(-1)(\lambda+7) - (3)(\lambda-2)] - \lambda[(2)(\lambda+7) - (3)(\lambda+3)] + (\lambda+1)[(2)(\lambda-2) - (-1)(\lambda+3)] <br> = (\lambda-1)[-\lambda-7-3\lambda+6] - \lambda[2\lambda+14-3\lambda-9] + (\lambda+1)[2\lambda-4+\lambda+3] <br> = (\lambda-1)(-4\lambda-1) - \lambda(-\lambda+5) + (\lambda+1)(3\lambda-1) <br> = (-4\lambda^2-\lambda+4\lambda+1) - (-\lambda^2+5\lambda) + (3\lambda^2-\lambda+3\lambda-1) <br> = -4\lambda^2+3\lambda+1 + \lambda^2-5\lambda + 3\lambda^2+2\lambda-1 <br> = (-4+1+3)\lambda^2 + (3-5+2)\lambda + (1-1) <br> = 0\lambda^2 + 0\lambda + 0 = 0. <br> Since the determinant is 0 for all real values of \lambda, there are infinite real values of \lambda for which the matrix has no inverse.
Correct Answer: 4