University of Regina MATH 122 - Linear Algebra I
MATH 122 - Linear Algebra I
§2.3. Properties of matrix multiplication
Martin Frankland
October 5, 2022
Example 1 (#2.3.6). Let A be a 2 × 2 matrix.
0 1 a b
(a) If A commutes with , show that A = holds for some a, b ∈ R.
0 0 0 a
Solution. Write the entries of A as
a b
A= .
c d
Let us compute the matrix product in the two different orders:
0 1 a b 0 1
A =
0 0 c d 0 0
0 a
=
0 c
0 1 0 1 a b
A=
0 0 0 0 c d
c d
= .
0 0
Setting the two products to be equal yields the equation
0 a c d
=
0 c 0 0
(
c=0
⇐⇒
a=d
a b
⇐⇒ A =
0 a
where a, b ∈ R are arbitrary.
© 2022 Martin Frankland All Rights Reserved 1
MATH 122 - Linear Algebra I
§2.3. Properties of matrix multiplication
Martin Frankland
October 5, 2022
Example 1 (#2.3.6). Let A be a 2 × 2 matrix.
0 1 a b
(a) If A commutes with , show that A = holds for some a, b ∈ R.
0 0 0 a
Solution. Write the entries of A as
a b
A= .
c d
Let us compute the matrix product in the two different orders:
0 1 a b 0 1
A =
0 0 c d 0 0
0 a
=
0 c
0 1 0 1 a b
A=
0 0 0 0 c d
c d
= .
0 0
Setting the two products to be equal yields the equation
0 a c d
=
0 c 0 0
(
c=0
⇐⇒
a=d
a b
⇐⇒ A =
0 a
where a, b ∈ R are arbitrary.
© 2022 Martin Frankland All Rights Reserved 1
