1.
\(\displaystyle
\begin{bmatrix}
7 \amp 0 \amp 0 \\
0 \amp 7 \amp 0 \\
0 \amp 0 \amp 7
\end{bmatrix}
-
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}\)
Exercises 7.6 Exercises
Matrix operations involving special forms.
Carry out each matrix operation using the patterns discovered in this chapter. Reflect on how the patterns in the entries of the matrices involved affect the patterns in the entries of the result.
2.
\(\displaystyle
7
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}\)
3.
\(\displaystyle
4
\begin{bmatrix}
7 \amp 0 \amp 0 \\
0 \amp 7 \amp 0 \\
0 \amp 0 \amp 7
\end{bmatrix}\)
4.
\(\displaystyle
\begin{bmatrix}
7 \amp 0 \amp 0 \\
0 \amp 7 \amp 0 \\
0 \amp 0 \amp 7
\end{bmatrix}
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}\)
5.
\(\displaystyle
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}
\begin{bmatrix}
7 \amp 0 \amp 0 \\
0 \amp 7 \amp 0 \\
0 \amp 0 \amp 7
\end{bmatrix}\)
6.
\(\displaystyle
{
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}
}^3\)
7.
\(\displaystyle
4
{
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}
}^2\)
8.
\(\displaystyle
\utrans{
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}
}\)
9.
\(\displaystyle
\utrans{
\begin{bmatrix}
1 \amp 0 \amp 0 \\
0 \amp 1 \amp 0 \\
0 \amp 0 \amp 1
\end{bmatrix}
}\)
10.
\(\displaystyle
\utrans{
\begin{bmatrix}
0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0
\end{bmatrix}
}\)
11.
\(\displaystyle
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}
+
\begin{abmatrix}{rrr}
4 \amp 0 \amp 0 \\
0 \amp -2 \amp 0 \\
0 \amp 0 \amp -3
\end{abmatrix}\)
12.
\(\displaystyle
-3
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}\)
13.
\(\displaystyle
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}
\begin{bmatrix}
\frac{1}{5} \amp 0 \amp 0 \\
0 \amp 3 \amp 0 \\
0 \amp 0 \amp 2
\end{bmatrix}\)
14.
\(\displaystyle
\begin{bmatrix}
\frac{1}{5} \amp 0 \amp 0 \\
0 \amp 3 \amp 0 \\
0 \amp 0 \amp 2
\end{bmatrix}
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}\)
15.
\(\displaystyle
{
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \amp 0 \\
0 \amp 0 \amp 6 \amp 0 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}
}^5\)
16.
\(\displaystyle
\utrans{
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \amp 0 \\
0 \amp 0 \amp 6 \amp 0 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}
}\)
17.
\(\displaystyle
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}
+
\begin{abmatrix}{rrr}
0 \amp 4 \amp -2 \\
0 \amp 0 \amp 1 \\
0 \amp 0 \amp 0
\end{abmatrix}\)
18.
\(\displaystyle
\begin{abmatrix}{rrr}
-3 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \\
0 \amp 0 \amp 4
\end{abmatrix}
\begin{abmatrix}{rrr}
-2 \amp 6 \amp -3 \\
3 \amp -3 \amp 7 \\
-1 \amp 8 \amp -7
\end{abmatrix}\)
19.
\(\displaystyle
\begin{abmatrix}{rrr}
-2 \amp 6 \amp -3 \\
3 \amp -3 \amp 7 \\
-1 \amp 8 \amp -7
\end{abmatrix}
\begin{abmatrix}{rrr}
-3 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \\
0 \amp 0 \amp 4
\end{abmatrix}\)
20.
\(\displaystyle
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}
+
\begin{abmatrix}{rrr}
6 \amp 1 \amp 5 \\
0 \amp -3 \amp -3 \\
0 \amp 0 \amp 4
\end{abmatrix}\)
21.
\(\displaystyle
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
5 \amp -2 \amp 0 \\
6 \amp 2 \amp 5
\end{abmatrix}
-
\begin{abmatrix}{rrr}
-5 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \\
-1 \amp 2 \amp 4
\end{abmatrix}\)
22.
\(\displaystyle
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
5 \amp -2 \amp 0 \\
6 \amp 2 \amp 5
\end{abmatrix}
-
\begin{bmatrix}
0 \amp 0 \amp 0 \\
5 \amp 0 \amp 0 \\
6 \amp 2 \amp 0
\end{bmatrix}\)
23.
\(\displaystyle
5
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}\)
24.
\(\displaystyle
-4
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
-5 \amp 1 \amp 0 \\
4 \amp 0 \amp -2
\end{abmatrix}\)
25.
\(\displaystyle
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
5 \amp -2 \amp 0 \\
6 \amp 2 \amp 5
\end{abmatrix}
\begin{abmatrix}{rrr}
-5 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \\
-1 \amp 2 \amp 4
\end{abmatrix}\)
26.
\(\displaystyle
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}
\begin{abmatrix}{rrr}
6 \amp 1 \amp 5 \\
0 \amp -3 \amp -3 \\
0 \amp 0 \amp 4
\end{abmatrix}\)
27.
\(\displaystyle
{
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}
}^2\)
28.
\(\displaystyle
{
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
5 \amp -2 \amp 0 \\
6 \amp 2 \amp 5
\end{abmatrix}
}^2\)
29.
\(\displaystyle
\utrans{
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}
}\)
30.
\(\displaystyle
\utrans{
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
-5 \amp 1 \amp 0 \\
4 \amp 0 \amp -2
\end{abmatrix}
}\)
31.
\(\displaystyle
-2
\begin{abmatrix}{rrr}
1 \amp 2 \amp 3 \\
2 \amp 4 \amp -1 \\
3 \amp -1 \amp 5
\end{abmatrix}\)
32.
\(\displaystyle
\utrans{
\begin{abmatrix}{rrr}
7 \amp -6 \amp 0 \\
-6 \amp 4 \amp 1 \\
0 \amp 1 \amp 5
\end{abmatrix}
}\)
33.
\(\displaystyle
\begin{abmatrix}{rrr}
-4 \amp 1 \amp -2 \\
1 \amp 2 \amp -3 \\
-2 \amp -3 \amp 3
\end{abmatrix}
-
\utrans{
\begin{abmatrix}{rrr}
-4 \amp 1 \amp -2 \\
1 \amp 2 \amp -3 \\
-2 \amp -3 \amp 3
\end{abmatrix}
}\)
34.
\(\displaystyle
\begin{bmatrix}
0 \amp 0 \amp 0 \\
5 \amp 0 \amp 0 \\
6 \amp 2 \amp 0
\end{bmatrix}
+ \utrans{
\begin{bmatrix}
0 \amp 0 \amp 0 \\
5 \amp 0 \amp 0 \\
6 \amp 2 \amp 0
\end{bmatrix}
}\)
Inverses of special forms.
Compute the inverse of each matrix (if possible), either by inspection or using Procedure 6.3.7. If you use Procedure 6.3.7, afterward reflect on how the row operations involved guarantee that the result will be the of the same form as the original.
35.
\(\displaystyle
\begin{bmatrix}
3 \amp 0 \amp 0 \\
0 \amp 3 \amp 0 \\
0 \amp 0 \amp 3
\end{bmatrix}\)
36.
\(\displaystyle
\begin{abmatrix}{cccr}
4 \amp 0 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \amp 0 \\
0 \amp 0 \amp -\frac{1}{6} \amp 0 \\
0 \amp 0 \amp 0 \amp -7
\end{abmatrix}\)
37.
\(\displaystyle
\begin{abmatrix}{cccr}
4 \amp 0 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \amp -7
\end{abmatrix}\)
38.
\(\displaystyle
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 2 \amp 1 \\
0 \amp 1 \amp -3 \amp -1 \\
0 \amp 0 \amp -4 \amp -6 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}\)
39.
\(\displaystyle
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 2 \amp 1 \\
0 \amp 0 \amp -3 \amp -1 \\
0 \amp 0 \amp -4 \amp -6 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}\)
40.
\(\displaystyle
\begin{abmatrix}{rrrr}
4 \amp 0 \amp 0 \amp 0 \\
-4 \amp 2 \amp 0 \amp 0 \\
3 \amp 6 \amp 2 \amp 0 \\
-3 \amp 2 \amp 6 \amp -4
\end{abmatrix}\)
41.
\(\displaystyle
\begin{abmatrix}{rrrr}
4 \amp 0 \amp 0 \amp 0 \\
-4 \amp 2 \amp 0 \amp 0 \\
3 \amp 6 \amp 0 \amp 0 \\
-3 \amp 2 \amp 6 \amp -4
\end{abmatrix}\)
42. Operation patterns involving special forms.
Verify all parts of Proposition 7.5.1, not through examples, but by constructing a convincing, general argument based on the pattern of entries in the matrices involved, without assuming a specific size of matrix. Refer to the two provided partial proofs of Proposition 7.5.1 as examples. You may also wish to refer to your work on Discovery 7.5 and Discovery 7.6, as well as some of your answers to Exercises 4.6.102–4.6.108.
Below, the statements of Proposition 7.5.1 have been separated but with the form of matrix left blank — you should verify each statement once for each of the different forms of scalar, diagonal, upper triangular, lower triangular, and symmetric.
(a)
The result of adding two matrices is always a matrix.
(b)
The result of scalar multiplying a matrix is always a matrix.
(c)
The result of multiplying two matrices is always a matrix.
Note. Omit this statement for symmetric matrices.
(d)
The result of taking an inverse of an invertible matrix is always a matrix.
(e)
The result of taking a power (with positive or negative exponent) of a matrix is always a matrix.
43. Products of symmetric matrices.
Statement 3 of Proposition 7.5.1 explicitly states that it is not true in general that the product of two symmetric matrices is guaranteed to also be symmetric. However, that does not mean it can never happen.
(a)
Devise an example of two symmetric \(2 \times 2\) matrices whose product is not symmetric.
(b)
Devise an example of two symmetric \(2 \times 2\) matrices whose product is symmetric.
(c)
Prove that if two symmetric matrices of the same size commute (that is, produce the same result when multiplied in either order), then their product must also be symmetric.
(d)
Prove that the condition of Task c is the only way that symmetric matrices produce a symmetric product. That is, prove that if \(A\) and \(B\) are symmetric matrices of the same size so that the product \(A B\) is also symmetric, then the the two matrices must commute (that is, it must be true that \(B A = A B\)).
Skew-symmetric matrices.
A square matrix \(A\) is called skew-symmetric if \(\utrans{A} = -A\text{.}\) (Compare with the definition of symmetric matrix.)
The remaining exercises concern this new definition.
44. Patterns.
(a)
What patterns in the entries \(a_{ij}\) of a matrix \(A\) does the condition \(\utrans{A} = -A\) imply?
(b)
What further can you say about the diagonal entries of a skew-symmetric matrix, based on the general pattern you identified in Task a?
45. Examples.
Write down nonzero examples of skew-symmetric matrices of size \(2 \times 2\text{,}\) \(3 \times 3\text{,}\) and \(4 \times 4\text{.}\)
46. Properties.
(a)
Verify each of the following statements using general arguments in the same manner as in Exercise 7.6.42.
-
The result of adding two skew-symmetric matrices is always a skew-symmetric matrix.
-
The result of scalar multiplying a skew-symmetric matrix is always a skew-symmetric matrix.
(b)
If a skew-symmetric matrix is also invertible, must its inverse also be skew-symmetric? If the answer is yes, provide a general argument verifying your claim. If the answer is no, provide an example of a skew-symmetric matrix whose inverse is not skew-symmetric.
47. Symmetric/skew-symmetric decompositions.
(a)
(b)
Verify that the difference \(A - \utrans{A}\) is always skew-symmetric, for every square matrix \(A\text{.}\)
(c)
(d)
Apply your idea from Task c to the example matrices below. That is, for each matrix below, create two matrices, where the first is symmetric, the second is skew-symmetric, and the two sum to the original matrix.
-
\(\displaystyle \begin{abmatrix}{rr} 1 \amp -1 \\ -4 \amp 3 \end{abmatrix}\)
-
\(\displaystyle \begin{abmatrix}{rrr} 6 \amp 5 \amp 2 \\ 5 \amp 0 \amp -3 \\ -5 \amp 3 \amp 4 \end{abmatrix}\)
