A First Course in Linear Algebra, 2017a

1.2. Systems of Equations, Algebraic Procedures 49
Exercise 1.2.56 Find the rank of the following matrix.
⎡
⎢
⎣
−1 3 4 −3 8
1 −3 −4 2 −5
1 −3 −4 1 −2
−2 6 8 −2 4
⎤
⎥
⎦
Exercise 1.2.57 Suppose A is an m × n matrix. Explain why the rank of A is always no larger than
min(m,n).
Exercise 1.2.58 State whether each of the following sets of data are possible for the matrix equation
AX = B. If possible, describe the solution set. That is, tell whether there exists a unique solution, no
solution or infinitely many solutions. Here, [A|B] denotes the augmented matrix.
(a) A is a 5 × 6 matrix, rank(A)=4 and rank[A|B]=4.
(b) A is a 3 × 4 matrix, rank(A)=3 and rank[A|B]=2.
(c) A is a 4 × 2 matrix, rank(A)=4 and rank[A|B]=4.
(d) A is a 5 × 5 matrix, rank(A)=4 and rank[A|B]=5.
(e) A is a 4 × 2 matrix, rank(A)=2 and rank[A|B]=2.
Exercise 1.2.59 Consider the system −5x + 2y − z = 0 and −5x − 2y − z = 0. Both equations equal zero
and so −5x + 2y − z = −5x − 2y − z which is equivalent to y = 0. Does it follow that x and z can equal
anything? Notice that when x = 1, z= −4, and y = 0 are plugged in to the equations, the equations do
not equal 0. Why?
Exercise 1.2.60 Balance the following chemical reactions.
(a) KNO 3 + H 2 CO 3 → K 2 CO 3 + HNO 3
(b) AgI + Na 2 S → Ag 2 S + NaI
(c) Ba 3 N 2 + H 2 O → Ba(OH) 2 + NH 3
(d) CaCl 2 + Na 3 PO 4 → Ca 3 (PO 4 ) 2 + NaCl
Exercise 1.2.61 In the section on dimensionless variables it was observed that ρV 2 AB has the units of
force. Describe a systematic way to obtain such combinations of the variables which will yield something
which has the units of force.
Exercise 1.2.62 Consider the following diagram of four circuits.