If the system of equations $(\lambda-1)x+(\lambda-4)y+\lambda z=5$, $\lambda x+(\lambda-1)y+(\lambda-4)z=7$, $(\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$ has infinitely many solutions, then $\lambda^2+\lambda$ is equal to

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] The determinant of the coefficients is $-2(2\lambda+1)(\lambda-3)$, so $\lambda=3$ or $-\frac12$. For $\lambda=-\frac12$ the system is inconsistent. For $\lambda=3$ the third equation equals $-1	imes$(first)$+2	imes$(second) on both sides ($-5+14=9$), so it is consistent with infinitely many solutions. $\lambda^2+\lambda=12$. Answer (4).

Q. If the system of equations $(\lambda-1)x+(\lambda-4)y+\lambda z=5$, $\lambda x+(\lambda-1)y+(\lambda-4)z=7$, $(\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$ has infinitely many solutions, then $\lambda^2+\lambda$ is equal to

Explanation

Reviewed by Dr. Abubacker

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