Answer:
a) the number of equations=2 and the number of variables=2
b) the system is consistent for all k , such that k≠(-2)
Step-by-step explanation:
a) for the matrix
![A= \left[\begin{array}{ccc}1&k&3\\-2&4&1\end{array}\right] \\](https://tex.z-dn.net/?f=A%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26k%263%5C%5C-2%264%261%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
then the number of equations = number of rows = 2 , and the number of variables = number of columns - 1 (that correspond to the independent vector or results of the equation) = 3-1 = 2
b) the system is consistent when there is at least one solution. Then the system will be inconsistent for
![\left[\begin{array}{ccc}1*(-2)&k*(-2)&3*(-2)\\-2&4&1\end{array}\right] \\=\left[\begin{array}{ccc}-2&(-2k)&-6\\-2&4&1\end{array}\right] \\](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%2A%28-2%29%26k%2A%28-2%29%263%2A%28-2%29%5C%5C-2%264%261%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-2%26%28-2k%29%26-6%5C%5C-2%264%261%5Cend%7Barray%7D%5Cright%5D%20%5C%5C)
for -2*k=4 → k=(-2) , the system is not consistent since the same equation -2*x + 4*y =(-6) according to the first row , while in the second row -2*x + 4*y =1 (that would mean (-6)=1 )
then the system is consistent for all k , such that k≠(-2)
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