Page 33 - 35Linear Algebra

1.5 Review Problems 33

6. Matrix Multiplication: Let M and N be matrices

a b e f
M = and N = ,
c d g h
and v the vector

x
v = .
y
If we first apply N and then M to v we obtain the vector MNv.

(a) Show that the composition of matrices MN is also a linear oper-
ator.

(b) Write out the components of the matrix product MN in terms of
the components of M and the components of N. Hint: use the
general rule for multiplying a 2-vector by a 2×2 matrix.
(c) Try to answer the following common question, “Is there any sense
in which these rules for matrix multiplication are unavoidable, or
are they just a notation that could be replaced by some other
notation?”
(d) Generalize your multiplication rule to 3 × 3 matrices.

7. Diagonal matrices: A matrix M can be thought of as an array of num-
i
bers m , known as matrix entries, or matrix components, where i and j
j
index row and column numbers, respectively. Let

1 2 i
M = = m .
3 4 j
1
2
1
2
Compute m , m , m and m .
1 2 1 2
i
The matrix entries m whose row and column numbers are the same
i
i
are called the diagonal of M. Matrix entries m with i 6= j are called
j
off-diagonal. How many diagonal entries does an n × n matrix have?
How many off-diagonal entries does an n × n matrix have?
If all the off-diagonal entries of a matrix vanish, we say that the matrix
is diagonal. Let
0
λ 0 0 λ 0
D = and D = .
0 µ 0 µ 0
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