Linear Algebra question: If is the augmented matrix of a system of linear equati
ID: 3113373 • Letter: L
Question
Linear Algebra question:
If is the augmented matrix of a system of linear equations, determine the number of equations and the number of variables. (b) If is the augmented matrix of a system of linear equations, find the value(s) of such that the system is consistent. (c) If is the coefficient matrix of a homogeneous system of linear equations, determine the number of equations and the number of variables. (d) If is the coefficient matrix of a homogeneous system of linear equations, find the value(s) of such that the system is consistent
A = 2 -1 3
-4 2 k
4 -2 6
I tried to solve it, but would like to check my work, any help would be awesome.
Explanation / Answer
Ans(a):
if system of equation is represented by AX=B then it's augmented form is given by [A B].
that means last column of given matrix represents right side values of each equation.
given matrix A has 3 rows so that means there will be 3 equations.
given matrix A has 3 columns in which last column is for constant terms for right side of equation so are left with only two columns. That indicates there are only two variables (say x and y)
Ans(b):
using given augmented matrix we can write system of equation as:
2x-1y=3...(i)
-4x+2y=k...(ii)
4x-2y=6...(iii)
solve (i) and (ii)
2x-1y=3...(i)
-4x+2y=k...(ii)
4x-2y=6...(iii)
rewrite (i)
2x-1y=3
2x-3=y...(iv)
plug (iv) into (iii)
4x-2(2x-3)=6
4x-4x+6=6
6=6
which is true so system will have infinitely many solutions
so any value of x,y will satisfy the system of equations (i,ii,iii)
as values of x and y are not fixed to value of k is also not fixed
say x=0, y=0 then plugging into (ii) gives k=0
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since last column of given matrix is non zero, that means right side of each equation must be non zero while homogeneous system has all right side entry 0.
So that means given augmented matrix is not for homogeneous system and hence question number c and d are not applicable.
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