Composition and Matrix Multiplication
Exercise Compute the composition TS.
T(y) = 1  y^{2}:
R → R
S(x) = sin x:
R → R
T(x, y) = x^{2} + y^{2}:
R^{2} → R
S(t) = (cos t, sin t):
R → R^{2}
T(x, y) = (x + y, x  y):
R^{2} → R^{2}
S(u, v) = (1/2(u + v), 1/2(u  v)):
R^{2} → R^{2}
T(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})
= (x_{1}  2x_{2} + 3x_{3}  4x_{4} + 5x_{5}):
R^{5} → R
S(t) = (8t, 7t, 6t, 5t, 4t):
R → R^{5}
T(t) = (8t, 7t, 6t, 5t, 4t):
R → R^{5}
S(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})
= (x_{1}  2x_{2} + 3x_{3}  4x_{4} + 5x_{5}):
R^{5} → R
T(x_{1}, x_{2}, x_{3})
= (x_{1}  2x_{2} + 3x_{3},  2x_{1} + 3x_{2}  4x_{3}):
R^{3} → R^{2}
S(y_{1}, y_{2})
= ( 2y_{1}  y_{2}, y_{2}, y_{1} + y_{2}):
R^{2} → R^{3}
T(y_{1}, y_{2})
= ( 2y_{1}  y_{2}, y_{2}, y_{1} + y_{2}):
R^{2} → R^{3}
S(x_{1}, x_{2}, x_{3})
= (x_{1}  2x_{2} + 3x_{3},  2x_{1} + 3x_{2}  4x_{3}):
R^{3} → R^{2}
Answer
Exercise Draw the picture and identify the composition TS.
T = Reflection in xaxis: R^{2} → R^{2}
S = Reflection in yaxis: R^{2} → R^{2}
T = Rotation by 90 degrees: R^{2} → R^{2}
S = Reflection in origin: R^{2} → R^{2}
T = Reflection in origin: R^{2} → R^{2}
S = Rotation by 90 degrees: R^{2} → R^{2}
T = Reflection in xaxis: R^{2} → R^{2}
S = Embedding as diagonal: R → R^{2}
T = Embedding as diagonal: R → R^{2}
S = Projection onto xaxis: R^{2} → R
T = Projection onto xaxis: R^{2} → R
S = Embedding as diagonal: R → R^{2}
Answer
Exercise Compute matrix products.
[ 
1 
1 
][ 
1/2 
1/2 
] 
1 
1 
1/2 
1/2 
[ 

] 
[ 
8 
] 

7 
1 
2 
3 
4 
5 
6 

5 

4 
[ 
8 
] 
[ 

] 
7 

6 
1 
2 
3 
4 
5 
5 

4 

[ 
2 
1 
][ 
1 
2 
3 
] 
0 
1 
2 
3 
4 
[ 
1 
2 
3 
4 
] 
5 
6 
7 
8 
9 
10 
11 
12 


[ 
1 
1 
1 
][ 
1 
2 
3 
4 
] 
1 
2 
1 
5 
6 
7 
8 
2 
0 
1 
9 
10 
11 
12 
Answer
Exercise
For each of the following matrix A,
find all the matrices X satisfying AX = XA.
Answer
Exercise
For each of the following matrix A,
find all the two column matrices X satisfying AX = O (the zero matrix).
Answer
Exercise
For each of the following matrix A,
find all the matrices X satisfying AX = I (the identity matrix).
Answer
Exercise Let
A = [ 
2 
0 
2 
1 
] 
1 
1 
1 
0 
0 
1 
2 
1 
For each of the following matrix B,
find a matrix X such that AX = B.
Answer
Exercise Let
A = [ 
1 
1 
1 
0 
2 
],
B = [ 
1 
3 
1 
] 
2 
2 
0 
2 
2 
0 
2 
0 
1 
1 
2 
3 
1 
0 
3 
2 
2 
2 
1 
3 
1 
0 
0 
1 
Can you find a matrix X such that AX = B?
In general, what is the criterion for the existence of X
such that AX = B? What about the uniqueness of X?
Answer
Exercise Describe the first two rows and the last three columns of AB in terms of row and columns of
A and B.
Answer
Exercise The trace trA of a square matrix is the sum of
its diagonal entries. For example,
tr[ 
1 
2 
] = 1 + 4 = 5 
3 
4 
tr[ 
0 
3 
4 
] = 0 + 5  3 = 2 
2 
5 
7 
0 
4 
3 
Prove the trace has the following properties
tr(A + B) = trA + trB, tr cA = c trA, trA^{T} = trA, trAB = trBA.
Answer
Exercise What does the multiplication of two
upper triangular matrices look like? What about
lower triangular matrices?
diagonal matrices?
Answer
Exercise What are the matrices A satisfying
AX = XA for any X?
Answer
Exercise
 Prove that transformations T and S are onto ⇒
The composition TS is onto.
 What about the converse? In other words, if the composition TS is onto,
are the transformations T and S onto?
 Rephrase your conclusions in the first and second part in terms of matrix products.
Answer
Exercise
 Prove that transformations T and S are onotoone ⇒
The composition TS is onetoone.
 What about the converse? In other words, if the composition TS is onetoone,
are the transformations T and S onetoone?
 Rephrase your conclusions in the first and second part in terms of matrix products.
Answer