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Find the conical form of a quadratic form Q(x,y)=2x^2+2y^2-2xy by using an orthogonal transformation hence find nature,rank,index and signature of the conical form?
consider the following system of equations:
3x + 4y +5z = 66
7x + 4y +3z = 74
8x + 8y +9z = 136
a. write down the associated augmented matrix for this system of equations and the coefficient matrix A.
b. by performing elementary row operations of the augmented matrix, solve the system of equations or show that no solution exists. In case there exist infinitely many solutions, then the solution to the system must be written in parametric form.
c. Based on your answer in b, what is the rank of the coefficient matrix A?
Find the product of eigen value of matrix
Let A, B be two subrings of a ring R such that for all a e A, b e B,
ba e A then show that
(1) A + B is a subring of R.
(2) A is an ideal of A + B.
3) An B is an ideal of B.
Reduce the quadratic form 2𝑥
2 + 2𝑦
2 + 2𝑧
2 + 2𝑦𝑧 to the canonical form by
orthogonal reduction. Find the index, signature and nature of the quadratic form.
Express v = t2+4t-3 in P(t) as a linear combination of the
polynomials p =t2−2t+5,p =2t2 −3tand p =t+1.
Consider the following system of equations:
1)
x + 3y + 5z = b1
2x + 4y + 6z = b2.
Show that there are no solutions if b2 = 2b1 and infinitely many solutions otherwise.
2)
x + 4y = b1
2x + 5y = b2
3x + 6y = b3.
Show that there is exactly one solution if b3 = 7b1 − 2b2 and no solutions otherwise.
3. A firm can produce a good either by (i) a labor intensive technique, using 8 units of labor and 1 unit of capital or (ii) a capital intensive technique using 1 unit of labor and 2 units of capital. The firm can arrange up to 200 units of labor and 100 units of capital. Note that the firm produce goods X and Y in process land 1 and 2 respectively. Its objective is to maximize profit by selling the good.
i) Construct the objective function and the constant inequalities.
ii) By drawing the graphs of the linear constraints, find the optimal solutions.
iii) Find the solution using the simplex methods.
. Work Out
1. Solve by crammer's rule, the following equations.
3X + 3Y - Z = 11
2X - Y + 2Z = 9
4X + 3Y - 27 = 25
2. solve the above equations using graphical method.
Define the following terms
PART 1.
a) What is an equation?
b) What is a linear equation?
c) What is determinants?
d) What is a matrix?
e) What is linear programming?
#339914 on Dec 2023