1. Use Gaussian elimination to determine if each of the following linear systems is consistent. Then use Gauss-Jordan elimination to solve the consistent linear systems. Use Notion 1.4.9 to denote the elementary row operations that you used for each step.
a. 2x - 2y + 4z = 5
x + 3y + 6z = 2
x + 7y + 10z = 1.
b. x - 2y + z = 4
2x + 6y + 3z = 5
-x + 2y - 7z = 2.
c. x1 + 3x2 - x3 + 4x4 - x5 = 2
2x1 + x2 + 4x3 - 3x4 + 5x5 = 1
-x1 + 2x2 + 3x3 - x4 + 2x5 = 9.
2. Determine the values of λ so that the following homogenous linear system has only the trivial solution:
λx + y + z = 0
x + λy + z = 0
x + y + λz = 0.
3. For which real numbers a and b does the following linear system have (a) no solution, (b) exactly one solution, or (c) infinitely many solutions? Justify your answers.
(a - 1)x + (a + 3)y + z = 1
ax + a2y + z = 1
x + ay + z = b.
4. A certain library owns 10 000 books. Each month 20% of the books in the library are lent out and 80% of the books lent out are returned, while 10% remain lent out and 10% are reported lost. Finally, 25% of the books listed as lost the previous month are found and returned to the library. At present all the books are in the library. How many books will be in the library, lent out, and lost after three months?
5. Consider an economy with three industries: coal-mining operation, oil-drilling operation and electricity-generating plant.
i) To produce $1.00 of coal, the mining operation must purchase $0.10 of its own production, $0.40 of oil and $0.10 of electricity.
ii) To produce $1.00 of oil, it takes $0.20 of coal, $0.20 of oil and $0.20 of electricity.
iii) To produce $1.00 electricity, the electricity-generating plant must purchase $0.20 of coal, $0.40 of oil and consume $0.20 of electricity.
Assume that during one week, the economy has an exterior demand of $50,000 worth of coal, $75,000 worth of oil, and $100,000 worth of electricity. Find the production of each of the three industries in that week to exactly satisfy both the internal and the external demands.
6. For any real number θ, define the matrix Aθ =
(cos θ - sin θ)
(sin θ cos θ)
(i) Prove that Aθ1Aθ2 = Aθ1+θ2 .
(ii) Let B = ( 1 0 0 -1) . Prove that BAθ = A-θB.
7. Let A and B be diagonal matrices of order n whose diagonal entries are
a1, a2, . . . , an and b1, b2, . . . , bn,
respectively. Prove that AB is the diagonal matrix whose diagonal entries are
a1b1, a2b2, . . . , anbn.
[Hint: Let A = (aij )n×n. Then aij = ai if i = j, and aij = 0 if i 6= j.]
8. Let A = (aij )n×n such that AB = BA for every square matrix B of order n.
(i) Prove that A is a diagonal matrix.
[Hint: Let Bi denote the matrix whose (i, i)-entry is 1 and 0 elsewhere. Show that aij = 0 whenever i 6= j.]
(ii) Prove that A is a scalar matrix.
[Hint: For i 6= j, let Bij denote the matrix whose (i, j)-entry is 1 and 0 elsewhere. Show that aii = ajj for all i 6= j.]