1.6.E: Operations with Matrices (Exercises)

Exercise \(\PageIndex{1}\)

Let \(M=\left[\begin{array}{rr}
2 & 3 \\
-2 & 1 \\
4 & -1
\end{array}\right]\) and \(N=\left[\begin{array}{rr}
3 & -2 \\
1 & 0 \\
2 & -5
\end{array}\right] \). Evaluate the following.

(a) \(3M\)

(b) \(M-N\)

(c) \(2 M+N\)

(d) \(2 N-6 M\)

Answer

(a) \(\left[\begin{array}{rr}
6 & 9 \\
-6 & 3 \\
12 & -3
\end{array}\right]\)

(b) \(\left[\begin{array}{rr}
-1 & 5 \\
-3 & 1 \\
2 & 4
\end{array}\right]\)

(c) \(\left[\begin{array}{rr}
7 & 4 \\
-3 & 2 \\
10 & -7
\end{array}\right]\)

(d) \(\left[\begin{array}{cc}
-14 & 18 \\
-10 & 2 \\
-4 & 28
\end{array}\right]\)

Exercise \(\PageIndex{2}\)

Evaluate the following matrix products.

(a) \(\left[\begin{array}{rr}
3 & 2 \\
-1 & 1
\end{array}\right]\left[\begin{array}{l}
2 \\
3
\end{array}\right]\)

(b) \(\left[\begin{array}{rr}
2 & -3 \\
1 & 4
\end{array}\right]\left[\begin{array}{rr}
1 & 4 \\
2 & -2
\end{array}\right]\)

(c) \(\left[\begin{array}{rrr}
2 & 1 & 3 \\
-3 & 2 & 1
\end{array}\right]\left[\begin{array}{rrr}
3 & 4 & -1 \\
0 & 2 & 4 \\
2 & 1 & -2
\end{array}\right]\)

(d) \(\left[\begin{array}{llll}
1 & 2 & 3 & -1
\end{array}\right]\left[\begin{array}{rr}
2 & 1 \\
3 & 1 \\
-2 & 4 \\
0 & -4
\end{array}\right]\)

Answer

(a) \(\left[\begin{array}{c}
12 \\
1
\end{array}\right]\)

(b) \(\left[\begin{array}{rr}
-4 & 14 \\
9 & -4
\end{array}\right]\)

(c) \(\left[\begin{array}{ccr}
12 & 13 & -4 \\
-7 & -7 & 9
\end{array}\right]\)

(d) \(\left[\begin{array}{ll}
2 & 19
\end{array}\right]\)

Exercise \(\PageIndex{3}\)

Suppose \(L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) and \(K: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) are defined by

\[ L(x, y, z)=(2 x+3 y, y-x+2 z, x+2 y-z) \nonumber \]

and

\[ K(x, y, z)=(2 x+4 y-3 z, x+y+z, 3 x-y+4 z). \nonumber \]

Find the matrices for the following linear functions.

(a) \(3 L\)

(b) \(L+K\)

(c) \(2 L-K\)

(d) \(K+2 L\)

(e) \(K \circ L\)

(f) \(L \circ K\)

Answer

(a) \(\left[\begin{array}{rrr}
6 & 9 & 0 \\
-3 & 3 & 6 \\
3 & 6 & -3
\end{array}\right]\)

(b) \(\left[\begin{array}{rrr}
4 & 7 & -3 \\
0 & 2 & 3 \\
4 & 1 & 3
\end{array}\right]\)

(c) \(\left[\begin{array}{rrr}
2 & 2 & 3 \\
-3 & 1 & 3 \\
-1 & 5 & -6
\end{array}\right]\)

(d) \(\left[\begin{array}{rcr}
6 & 10 & -3 \\
-1 & 3 & 5 \\
5 & 3 & 2
\end{array}\right]\)

(e) \(\left[\begin{array}{ccc}
-3 & 4 & 11 \\
2 & 6 & 1 \\
11 & 16 & -6
\end{array}\right]\)

(f) \(\left[\begin{array}{ccc}
7 & 11 & -3 \\
5 & -5 & 12 \\
1 & 7 & -5
\end{array}\right]\)

Exercise \(\PageIndex{4}\)

Let \(\mathbb{R}_{\theta}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear function which rotates a vector in \(\mathbb{R}^{2}\) counterclockwise through an angle θ. In Section 1.5 we saw that

\[ R_{\theta}(x, y)=\left[\begin{array}{rr}
\cos (\theta) & -\sin (\theta) \\
\sin (\theta) & \cos (\theta)
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right] . \nonumber \]

Show that the matrix for \(R_{\theta} \circ R_{\alpha}\) is the same as the matrix for \(R_{\theta+\alpha}\). In other words, show that \(R_{\theta} \circ R_{\alpha}=R_{\theta+\alpha}\).

Exercise \(\PageIndex{5}\)

Compute the determinants of the following matrices.

(a) \(\left[\begin{array}{ll}
2 & 3 \\
1 & 4
\end{array}\right]\)

(b) \(\left[\begin{array}{rr}
-3 & -2 \\
1 & 2
\end{array}\right]\)

(c) \(\left[\begin{array}{rrr}
2 & 3 & 1 \\
1 & 2 & 9 \\
5 & -3 & -1
\end{array}\right]\)

(d) \(\left[\begin{array}{rrr}
-1 & 2 & -1 \\
3 & 1 & 0 \\
5 & -4 & 0
\end{array}\right]\)

(e) \(\left[\begin{array}{rrrr}
1 & 2 & -1 & 3 \\
4 & 3 & -2 & 1 \\
1 & 4 & -4 & 3 \\
1 & 3 & 3 & 1
\end{array}\right]\)

(f) \(\left[\begin{array}{rrrrr}
1 & 2 & -2 & 3 & 1 \\
0 & 2 & 0 & 2 & 0 \\
-3 & 2 & 0 & 1 & 5 \\
1 & 5 & -2 & 1 & 0 \\
6 & -5 & 0 & 2 & -4
\end{array}\right] \)

Answer

(a) 5

(b) -4

(c) 175

(d) 17

(e) -143

(f) 300

Exercise \(\PageIndex{6}\)

Find the area of the parallelogram in \(\mathbb{R}^{2}\) with vertices at (1,−2), (3,−1), (4,1), and (2,0).

Exercise \(\PageIndex{7}\)

Find the volume of the parallelepiped in \(\mathbb{R}^{3}\) with bottom vertices at (1,1,1), (2,3,2), (−1,4,3), and (−2,2,2) and top vertices at (1,0,5), (2,2,6), (−1,3,7), and (−2,1,6).

Answer

32

Exercise \(\PageIndex{8}\)

Let \(P\) be the 4-dimensional parallelepiped with adjacent edges \(\mathbf{a}_{1}=(2,1,2,1)\), \(\mathbf{a}_{2}=(-2,0,1,1)\), \(\mathbf{a}_{3}=(1,1,3,6)\), and \(\mathbf{a}_{4}=(-3,1,5,0)\). Find the volume of \(P\).

Answer

8

Exercise \(\PageIndex{9}\)

Find \(2 \times 2\) matrices \(A\) and \(B\) for which \(A B \neq B A\).

Exercise \(\PageIndex{10}\)

Verify that (1.6.25) and (1.6.26) hold for all \(2 \times 2\) and \(3 \times 3\) matrices.

Exercise \(\PageIndex{11}\)

An \(n \times n\) matrix \(M=\left[a_{i j}\right]\) is called a diagonal matrix if \(a_{i j}=0\) for all \(i \neq j\). Show that if \(M\) is a diagonal matrix, then \(\operatorname{det}(M)=a_{11} a_{22} \cdots a_{n n}\).

Exercise \(\PageIndex{12}\)

If \(M\) is an \(n \times m\) matrix, then the \(m \times n\) matrix \(M^{T}\) whose columns are the rows of \(M\) is called the transpose of \(M\). For example, if

\[ M=\left[\begin{array}{ll}
1 & 2 \\
3 & 4 \\
5 & 6
\end{array}\right] , \nonumber \]

then

\[ M^{T}=\left[\begin{array}{lll}
1 & 3 & 5 \\
2 & 4 & 6
\end{array}\right] . \nonumber \]

(a) Show that for a \(2 \times 2\) matrix \(M\), \(\operatorname{det}\left(M^{T}\right)=\operatorname{det}(M)\).

(b) Show that for a \(3 \times 3\) matrix \(M\), \(\operatorname{det}\left(M^{T}\right)=\operatorname{det}(M)\). (Hint: Using (1.6.26), expand \(\operatorname{det}(M)\) along the first row and \(\operatorname{det}\left(M^{T}\right)\) along the first column.)

(c) Use induction to show that for any \(n \times n\) matrix \(M\), \(\operatorname{det}\left(M^{T}\right)=\operatorname{det}(M)\). (Hint: Note that \(\left(M^{T}\right)_{i j}=\left(M_{j i}\right)^{T}\).)

Exercise \(\PageIndex{13}\)

Let \(\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)\) and \(\mathbf{y}=\left(y_{1}, y_{2}, y_{3}\right)\) be vectors in \(\mathbb{R}^{3}\) and let \(\mathbf{e}_{1}\), \(\mathbf{e}_{2}\), and \(\mathbf{e}_{3}\) be the standard basis vectors for \(\mathbb{R}^{3}\). Show that applying (1.6.20) to the array

\[ \left[\begin{array}{lll}
\mathbf{e}_{1} & \mathbf{e}_{2} & \mathbf{e}_{3} \\
x_{1} & x_{2} & x_{3} \\
y_{1} & y_{2} & y_{3}
\end{array}\right] \nonumber \]

yields \(\mathbf{x} \times \mathbf{y}\). Discuss what is correct and what is incorrect about the statement

\[ \mathbf{x} \times \mathbf{y}=\operatorname{det}\left[\begin{array}{lll}
\mathbf{e}_{1} & \mathbf{e}_{2} & \mathbf{e}_{3} \\
x_{1} & x_{2} & x_{3} \\
y_{1} & y_{2} & y_{3}
\end{array}\right] . \nonumber \]

Exercise \(\PageIndex{14}\)

Show that the set of all points \(\mathbf{x}=(x, y, z)\) in \(\mathbb{R}^{3}\) which satisfy the equation

\[ \operatorname{det}\left[\begin{array}{rrr}
x & y & z \\
1 & 2 & -1 \\
3 & 1 & 2
\end{array}\right]=0 \nonumber \]

is a plane passing through the points (0,0,0), (1,2,−1), and (3,1,2).

Answer

This is the set of all points which satisfy \(x-y-z=0\).

Exercise \(\PageIndex{15}\)

Verify directly that if \(L: \mathbb{R}^{m} \rightarrow \mathbb{R}^{p}\) and \(K: \mathbb{R}^{p} \rightarrow \mathbb{R}^{n}\) are linear functions, then \(K \circ L: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\) is also a linear function.