Question

Consider the following matrix.
A = [1 k 3
−2 4 1 ]
Assume that the matrix is the augmented matrix of a system of linear equations.
(a) Determine the number of equations and the number of variables.
(b) Find the value(s) of k such that the system is consistent.

Answer 1

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

$A= \left[\begin{array}{ccc}1&k&3\\-2&4&1\end{array}\right]$

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

$\left[\begin{array}{ccc}1*(-2)&k*(-2)&3*(-2)\\-2&4&1\end{array}\right] \\=\left[\begin{array}{ccc}-2&(-2k)&-6\\-2&4&1\end{array}\right]$

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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