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.
1.
\(\displaystyle
\begin{bmatrix}
7 \amp 0 \amp 0 \\
0 \amp 7 \amp 0 \\
0 \amp 0 \amp 7
\end{bmatrix}
-
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}\)
Answer.
\(\displaystyle
\begin{bmatrix}
3 \amp 0 \amp 0 \\
0 \amp 3 \amp 0 \\
0 \amp 0 \amp 3
\end{bmatrix}\)
2.
\(\displaystyle
7
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}\)
Answer.
\(\displaystyle
\begin{bmatrix}
28 \amp 0 \amp 0 \\
0 \amp 28 \amp 0 \\
0 \amp 0 \amp 28
\end{bmatrix}\)
3.
\(\displaystyle
\begin{bmatrix}
7 \amp 0 \amp 0 \\
0 \amp 7 \amp 0 \\
0 \amp 0 \amp 7
\end{bmatrix}
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}\)
Answer.
\(\displaystyle
\begin{bmatrix}
28 \amp 0 \amp 0 \\
0 \amp 28 \amp 0 \\
0 \amp 0 \amp 28
\end{bmatrix}\)
4.
\(\displaystyle
4
\begin{bmatrix}
7 \amp 0 \amp 0 \\
0 \amp 7 \amp 0 \\
0 \amp 0 \amp 7
\end{bmatrix}\)
Answer.
\(\displaystyle
\begin{bmatrix}
28 \amp 0 \amp 0 \\
0 \amp 28 \amp 0 \\
0 \amp 0 \amp 28
\end{bmatrix}\)
5.
\(\displaystyle
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}
\begin{bmatrix}
7 \amp 0 \amp 0 \\
0 \amp 7 \amp 0 \\
0 \amp 0 \amp 7
\end{bmatrix}\)
Answer.
\(\displaystyle
\begin{bmatrix}
28 \amp 0 \amp 0 \\
0 \amp 28 \amp 0 \\
0 \amp 0 \amp 28
\end{bmatrix}\)
6.
\(\displaystyle
{
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}
}^3\)
Answer.
\(\displaystyle
\begin{bmatrix}
64 \amp 0 \amp 0 \\
0 \amp 64 \amp 0 \\
0 \amp 0 \amp 64
\end{bmatrix}\)
7.
\(\displaystyle
4
{
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}
}^2\)
Answer.
\(\displaystyle
\begin{bmatrix}
64 \amp 0 \amp 0 \\
0 \amp 64 \amp 0 \\
0 \amp 0 \amp 64
\end{bmatrix}\)
8.
\(\displaystyle
\utrans{
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}
}\)
Answer.
\(\displaystyle
\begin{bmatrix}
4 \amp 0 \amp 0 \\
0 \amp 4 \amp 0 \\
0 \amp 0 \amp 4
\end{bmatrix}\)
9.
\(\displaystyle
\utrans{
\begin{bmatrix}
1 \amp 0 \amp 0 \\
0 \amp 1 \amp 0 \\
0 \amp 0 \amp 1
\end{bmatrix}
}\)
Answer.
\(\displaystyle
\begin{bmatrix}
1 \amp 0 \amp 0 \\
0 \amp 1 \amp 0 \\
0 \amp 0 \amp 1
\end{bmatrix}\)
10.
\(\displaystyle
\utrans{
\begin{bmatrix}
0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0
\end{bmatrix}
}\)
Answer.
\(\displaystyle
\begin{bmatrix}
0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0
\end{bmatrix}\)
11.
\(\displaystyle
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}
+
\begin{abmatrix}{rrr}
4 \amp 0 \amp 0 \\
0 \amp -2 \amp 0 \\
0 \amp 0 \amp -3
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
9 \amp 0 \amp 0 \\
0 \amp -3 \amp 0 \\
0 \amp 0 \amp 0
\end{abmatrix}\)
12.
\(\displaystyle
-3
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
-15 \amp 0 \amp 0 \\
0 \amp 3 \amp 0 \\
0 \amp 0 \amp -9
\end{abmatrix}\)
13.
\(\displaystyle
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}
\begin{bmatrix}
\frac{1}{5} \amp 0 \amp 0 \\
0 \amp 3 \amp 0 \\
0 \amp 0 \amp 2
\end{bmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
1 \amp 0 \amp 0 \\
0 \amp -3 \amp 0 \\
0 \amp 0 \amp 6
\end{abmatrix}\)
14.
\(\displaystyle
\begin{bmatrix}
\frac{1}{5} \amp 0 \amp 0 \\
0 \amp 3 \amp 0 \\
0 \amp 0 \amp 2
\end{bmatrix}
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
1 \amp 0 \amp 0 \\
0 \amp -3 \amp 0 \\
0 \amp 0 \amp 6
\end{abmatrix}\)
15.
\(\displaystyle
{
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \amp 0 \\
0 \amp 0 \amp 6 \amp 0 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}
}^5\)
Answer.
\(\displaystyle
\begin{abmatrix}{crcc}
32 \amp 0 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \amp 0 \\
0 \amp 0 \amp 7776 \amp 0 \\
0 \amp 0 \amp 0 \amp 243
\end{abmatrix}\)
16.
\(\displaystyle
\utrans{
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \amp 0 \\
0 \amp 0 \amp 6 \amp 0 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}
}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \amp 0 \\
0 \amp 0 \amp 6 \amp 0 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}\)
17.
\(\displaystyle
\begin{abmatrix}{rrr}
5 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 3
\end{abmatrix}
+
\begin{abmatrix}{rrr}
0 \amp 4 \amp -2 \\
0 \amp 0 \amp 1 \\
0 \amp 0 \amp 0
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
5 \amp 4 \amp -2 \\
0 \amp -1 \amp 1 \\
0 \amp 0 \amp 3
\end{abmatrix}\)
18.
\(\displaystyle
\begin{abmatrix}{rrr}
-3 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \\
0 \amp 0 \amp 4
\end{abmatrix}
\begin{abmatrix}{rrr}
-2 \amp 6 \amp -3 \\
3 \amp -3 \amp 7 \\
-1 \amp 8 \amp -7
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
6 \amp -18 \amp 9 \\
6 \amp - 6 \amp 14 \\
-4 \amp 32 \amp -28
\end{abmatrix}\)
19.
\(\displaystyle
\begin{abmatrix}{rrr}
-2 \amp 6 \amp -3 \\
3 \amp -3 \amp 7 \\
-1 \amp 8 \amp -7
\end{abmatrix}
\begin{abmatrix}{rrr}
-3 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \\
0 \amp 0 \amp 4
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
6 \amp 12 \amp -12 \\
-9 \amp -6 \amp 28 \\
3 \amp 16 \amp -28
\end{abmatrix}\)
20.
\(\displaystyle
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}
+
\begin{abmatrix}{rrr}
6 \amp 1 \amp 5 \\
0 \amp -3 \amp -3 \\
0 \amp 0 \amp 4
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
7 \amp -1 \amp 0 \\
0 \amp 3 \amp 2 \\
0 \amp 0 \amp 11
\end{abmatrix}\)
21.
\(\displaystyle
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
5 \amp -2 \amp 0 \\
6 \amp 2 \amp 5
\end{abmatrix}
-
\begin{abmatrix}{rrr}
-5 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \\
-1 \amp 2 \amp 4
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
8 \amp 0 \amp 0 \\
5 \amp -4 \amp 0 \\
7 \amp 0 \amp 1
\end{abmatrix}\)
22.
\(\displaystyle
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
5 \amp -2 \amp 0 \\
6 \amp 2 \amp 5
\end{abmatrix}
-
\begin{bmatrix}
0 \amp 0 \amp 0 \\
5 \amp 0 \amp 0 \\
6 \amp 2 \amp 0
\end{bmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
0 \amp -2 \amp 0 \\
0 \amp 0 \amp 5
\end{abmatrix}\)
23.
\(\displaystyle
5
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
5 \amp -10 \amp -25 \\
0 \amp 30 \amp 25 \\
0 \amp 0 \amp 35
\end{abmatrix}\)
24.
\(\displaystyle
-4
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
-5 \amp 1 \amp 0 \\
4 \amp 0 \amp -2
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
-12 \amp 0 \amp 0 \\
20 \amp -4 \amp 0 \\
-16 \amp 0 \amp 8
\end{abmatrix}\)
25.
\(\displaystyle
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}
\begin{abmatrix}{rrr}
6 \amp 1 \amp 5 \\
0 \amp -3 \amp -3 \\
0 \amp 0 \amp 4
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
6 \amp 7 \amp -9 \\
0 \amp -18 \amp 2 \\
0 \amp 0 \amp 28
\end{abmatrix}\)
26.
\(\displaystyle
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
5 \amp -2 \amp 0 \\
6 \amp 2 \amp 5
\end{abmatrix}
\begin{abmatrix}{rrr}
-5 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \\
-1 \amp 2 \amp 4
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
-15 \amp 0 \amp 0 \\
-25 \amp -4 \amp 0 \\
-35 \amp 14 \amp 20
\end{abmatrix}\)
27.
\(\displaystyle
{
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}
}^2\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
1 \amp -14 \amp -50 \\
0 \amp 36 \amp 65 \\
0 \amp 0 \amp 49
\end{abmatrix}\)
28.
\(\displaystyle
{
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
5 \amp -2 \amp 0 \\
6 \amp 2 \amp 5
\end{abmatrix}
}^2\)
Answer.
\(\displaystyle
\begin{bmatrix}
9 \amp 0 \amp 0 \\
5 \amp 4 \amp 0 \\
58 \amp 6 \amp 25
\end{bmatrix}\)
29.
\(\displaystyle
\utrans{
\begin{abmatrix}{rrr}
1 \amp -2 \amp -5 \\
0 \amp 6 \amp 5 \\
0 \amp 0 \amp 7
\end{abmatrix}
}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
1 \amp 0 \amp 0 \\
-2 \amp 6 \amp 0 \\
-5 \amp 5 \amp 7
\end{abmatrix}\)
30.
\(\displaystyle
\utrans{
\begin{abmatrix}{rrr}
3 \amp 0 \amp 0 \\
-5 \amp 1 \amp 0 \\
4 \amp 0 \amp -2
\end{abmatrix}
}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
3 \amp -5 \amp 4 \\
0 \amp 1 \amp 0 \\
0 \amp 0 \amp -2
\end{abmatrix}\)
31.
\(\displaystyle
-2
\begin{abmatrix}{rrr}
1 \amp 2 \amp 3 \\
2 \amp 4 \amp -1 \\
3 \amp -1 \amp 5
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
-2 \amp -4 \amp - 6 \\
-4 \amp -8 \amp 2 \\
-6 \amp 2 \amp -10
\end{abmatrix}\)
32.
\(\displaystyle
\utrans{
\begin{abmatrix}{rrr}
7 \amp -6 \amp 0 \\
-6 \amp 4 \amp 1 \\
0 \amp 1 \amp 5
\end{abmatrix}
}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrr}
7 \amp -6 \amp 0 \\
-6 \amp 4 \amp 1 \\
0 \amp 1 \amp 5
\end{abmatrix}\)
33.
\(\displaystyle
\begin{abmatrix}{rrr}
-4 \amp 1 \amp -2 \\
1 \amp 2 \amp -3 \\
-2 \amp -3 \amp 3
\end{abmatrix}
-
\utrans{
\begin{abmatrix}{rrr}
-4 \amp 1 \amp -2 \\
1 \amp 2 \amp -3 \\
-2 \amp -3 \amp 3
\end{abmatrix}
}\)
Answer.
\(\displaystyle
\begin{bmatrix}
0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0
\end{bmatrix}\)
34.
\(\displaystyle
\begin{bmatrix}
0 \amp 0 \amp 0 \\
5 \amp 0 \amp 0 \\
6 \amp 2 \amp 0
\end{bmatrix}
+ \utrans{
\begin{bmatrix}
0 \amp 0 \amp 0 \\
5 \amp 0 \amp 0 \\
6 \amp 2 \amp 0
\end{bmatrix}
}\)
Answer.
\(\displaystyle
\begin{bmatrix}
0 \amp 5 \amp 6 \\
5 \amp 0 \amp 2 \\
6 \amp 2 \amp 0
\end{bmatrix}\)
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.
\(\displaystyle
\begin{bmatrix}
3 \amp 0 \amp 0 \\
0 \amp 3 \amp 0 \\
0 \amp 0 \amp 3
\end{bmatrix}\)
Answer.
\(\displaystyle
\begin{bmatrix}
\frac{1}{3} \amp 0 \amp 0 \\
0 \amp \frac{1}{3} \amp 0 \\
0 \amp 0 \amp \frac{1}{3}
\end{bmatrix}\)
36.
\(\displaystyle
\begin{abmatrix}{cccr}
4 \amp 0 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \amp 0 \\
0 \amp 0 \amp -\frac{1}{6} \amp 0 \\
0 \amp 0 \amp 0 \amp -7
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{ccrc}
\frac{1}{4} \amp 0 \amp 0 \amp 0 \\
0 \amp \frac{1}{2} \amp 0 \amp 0 \\
0 \amp 0 \amp -6 \amp 0 \\
0 \amp 0 \amp 0 \amp -\frac{1}{7}
\end{abmatrix}\)
37.
\(\displaystyle
\begin{abmatrix}{cccr}
4 \amp 0 \amp 0 \amp 0 \\
0 \amp 2 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \amp -7
\end{abmatrix}\)
Answer.
Not invertible.
38.
\(\displaystyle
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 2 \amp 1 \\
0 \amp 1 \amp -3 \amp -1 \\
0 \amp 0 \amp -4 \amp -6 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrrr}
\frac{1}{2} \amp 0 \amp \frac{1}{4} \amp \frac{1}{3} \\
0 \amp 1 \amp -\frac{3}{4} \amp -\frac{7}{6} \\
0 \amp 0 \amp -\frac{1}{4} \amp -\frac{1}{2} \\
0 \amp 0 \amp 0 \amp \frac{1}{3}
\end{abmatrix}\)
39.
\(\displaystyle
\begin{abmatrix}{rrrr}
2 \amp 0 \amp 2 \amp 1 \\
0 \amp 0 \amp -3 \amp -1 \\
0 \amp 0 \amp -4 \amp -6 \\
0 \amp 0 \amp 0 \amp 3
\end{abmatrix}\)
Answer.
Not invertible.
40.
\(\displaystyle
\begin{abmatrix}{rrrr}
4 \amp 0 \amp 0 \amp 0 \\
-4 \amp 2 \amp 0 \amp 0 \\
3 \amp 6 \amp 2 \amp 0 \\
-3 \amp 2 \amp 6 \amp -4
\end{abmatrix}\)
Answer.
\(\displaystyle
\begin{abmatrix}{rrrr}
\frac{ 1}{4} \amp 0 \amp 0 \amp 0 \\
\frac{ 1}{2} \amp \frac{1}{2} \amp 0 \amp 0 \\
-\frac{15}{8} \amp -\frac{3}{2} \amp \frac{1}{2} \amp 0 \\
-\frac{11}{4} \amp -2 \amp \frac{3}{4} \amp -\frac{1}{4}
\end{abmatrix}\)
41.
\(\displaystyle
\begin{abmatrix}{rrrr}
4 \amp 0 \amp 0 \amp 0 \\
-4 \amp 2 \amp 0 \amp 0 \\
3 \amp 6 \amp 0 \amp 0 \\
-3 \amp 2 \amp 6 \amp -4
\end{abmatrix}\)
Answer.
Not invertible.
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 4.6.87–4.6.93.
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 \(B A = A B\)).
