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Question

Consider the system of equations
ax + y + z = 1
x + ay + z = 1
x + y + az = 1, then which of the following statement(s) is/are correct?


A

if a = 2, then the system has unique solution

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B

if a = 1, then the system has infinite solutions

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C

if a = –2, then the system has no solution

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D

if a = 2, then the system has infinite solutions

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Solution

The correct options are
A

if a = 2, then the system has unique solution


B

if a = 1, then the system has infinite solutions


C

if a = –2, then the system has no solution


Δ=∣ ∣a111a111a∣ ∣=(a1)2(a+2)

Δx=∣ ∣1111a1a1a∣ ∣=(a1)2

Δy=∣ ∣a1111111a∣ ∣=(a1)2

Δz=∣ ∣a111a1111∣ ∣=(a1)2

Clearly, if a = 2, then Δ0 system is consistent with unique solution.
If a = 1, then all three equations are identical, so system has infinite solutions
If a = -2, then Δ=0 but Δx0,Δy0,Δz0.
Thus, system is inconsistent.


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