Question

# Consider the following system of linear equations  ⎡⎢⎣21−443−1212−8⎤⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣a57⎤⎥⎦ Number of values of a for which system has infinitely many solutions is

Solution

## For the system ⎡⎢⎣21−443−1212−8⎤⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣a57⎤⎥⎦ has infinitely many solutions  D=∣∣ ∣∣21−443−1212−8∣∣ ∣∣=0 D1=0, for all values of a as C2 and C3 are identical. D2=0⇒∣∣ ∣∣2a−445−1217−8∣∣ ∣∣=0 Applying R2→R2−3R1, R2→R3−2R1 ∣∣ ∣∣2a−4−25−3a0−37−2a0∣∣ ∣∣=0⇒a=15 Now, D3=0 ⇒∣∣ ∣∣21a435127∣∣ ∣∣=0 Applying R2→R2−2R1, R3→R3−0.5R1 ⇒∣∣ ∣∣21a015−2a01.57−0.5a∣∣ ∣∣=0⇒a=15 Hence, there is only one value of a for which the system has infinitetly many solution.

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