Saltar al contenido principal

# 1.6.E: Operaciones con Matrices (Ejercicios)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

Ejercicio$$\PageIndex{1}$$

Vamos\ (M=\ left [\ begin {array} {rr}
2 & 3\\
-2 & 1\\
4 & -1
\ end {array}\ right]\) y\ (N=\ left [\ begin {array} {rr}
3 & -2\\
1 & 0\\
2 & -5
\ end {array}\ right]\). Evalúe lo siguiente.

(a)$$3M$$

b)$$M-N$$

c)$$2 M+N$$

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

Contestar

(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]\)

Ejercicio$$\PageIndex{2}$$

Evaluar los siguientes productos de matriz.

(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]\)

Contestar

(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]\)

Ejercicio$$\PageIndex{3}$$

Supongamos$$L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$$ y$$K: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$$ están definidos por

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

y

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

Encuentre las matrices para las siguientes funciones lineales.

(a)$$3 L$$

b)$$L+K$$

c)$$2 L-K$$

(d)$$K+2 L$$

(e)$$K \circ L$$

f)$$L \circ K$$

Contestar

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

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

(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}\ derecha]\)

(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]\)

Ejercicio$$\PageIndex{4}$$

Let$$\mathbb{R}_{\theta}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$$ Ser la función lineal que gira un vector en$$\mathbb{R}^{2}$$ sentido antihorario a través de un ángulo θ. En la Sección 1.5 vimos que

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

Demostrar que la matriz para$$R_{\theta} \circ R_{\alpha}$$ es la misma que la matriz para$$R_{\theta+\alpha}$$. En otras palabras, demuéstralo$$R_{\theta} \circ R_{\alpha}=R_{\theta+\alpha}$$.

Ejercicio$$\PageIndex{5}$$

Compute los determinantes de las siguientes 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}\ derecha]\)

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

Contestar

a) 5

b) -4

c) 175

d) 17

e) -143

f) 300

Ejercicio$$\PageIndex{6}$$

Encuentra el área del paralelogramo en$$\mathbb{R}^{2}$$ con vértices en (1, −2), (3, −1), (4, 1) y (2, 0).

Ejercicio$$\PageIndex{7}$$

Encuentra el volumen del paralelepípedo$$\mathbb{R}^{3}$$ con los vértices inferiores en (1, 1, 1), (2, 3, 2), (−1, 4, 3) y (−2, 2, 2) y los vértices superiores en (1, 0, 5), (2 , 2, 6), (−1, 3, 7) y (−2, 1, 6).

Contestar

32

Ejercicio$$\PageIndex{8}$$

Dejar$$P$$ ser el paralelepípedo de 4 dimensiones con bordes adyacentes$$\mathbf{a}_{1}=(2,1,2,1)$$,$$\mathbf{a}_{2}=(-2,0,1,1)$$,$$\mathbf{a}_{3}=(1,1,3,6)$$, y$$\mathbf{a}_{4}=(-3,1,5,0)$$. Encuentra el volumen de$$P$$.

Contestar

8

Ejercicio$$\PageIndex{9}$$

Encuentra$$2 \times 2$$ matrices$$A$$ y$$B$$ para las cuales$$A B \neq B A$$.

Ejercicio$$\PageIndex{10}$$

Verificar que (1.6.25) y (1.6.26) se mantengan para todos$$2 \times 2$$ y$$3 \times 3$$ matrices.

Ejercicio$$\PageIndex{11}$$

Una$$n \times n$$ matriz$$M=\left[a_{i j}\right]$$ se llama matriz diagonal si es$$a_{i j}=0$$ para todos$$i \neq j$$. Mostrar que si$$M$$ es una matriz diagonal, entonces$$\operatorname{det}(M)=a_{11} a_{22} \cdots a_{n n}$$.

Ejercicio$$\PageIndex{12}$$

Si$$M$$ es una$$n \times m$$ matriz, entonces la$$m \times n$$ matriz$$M^{T}$$ cuyas columnas son las filas de$$M$$ se llama la transposición de$$M$$. Por ejemplo, si

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

entonces

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

(a) Demostrar que para una$$2 \times 2$$ matriz$$M$$,$$\operatorname{det}\left(M^{T}\right)=\operatorname{det}(M)$$.

(b) Demostrar que para una$$3 \times 3$$ matriz$$M$$,$$\operatorname{det}\left(M^{T}\right)=\operatorname{det}(M)$$. (Pista: Usando (1.6.26), expanda$$\operatorname{det}(M)$$ a lo largo de la primera fila y$$\operatorname{det}\left(M^{T}\right)$$ a lo largo de la primera columna.)

(c) Utilizar inducción para demostrar que para cualquier$$n \times n$$ matriz$$M$$,$$\operatorname{det}\left(M^{T}\right)=\operatorname{det}(M)$$. (Pista: Tenga en cuenta que$$\left(M^{T}\right)_{i j}=\left(M_{j i}\right)^{T}$$.)

Ejercicio$$\PageIndex{13}$$

Dejar$$\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)$$ y$$\mathbf{y}=\left(y_{1}, y_{2}, y_{3}\right)$$ ser vectores en$$\mathbb{R}^{3}$$ y dejar$$\mathbf{e}_{1}$$,$$\mathbf{e}_{2}$$, y$$\mathbf{e}_{3}$$ ser los vectores base estándar para$$\mathbb{R}^{3}$$. Mostrar que aplicar (1.6.20) a la matriz

\ [\ 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}\ nono]\ umber\]

rendimientos$$\mathbf{x} \times \mathbf{y}$$. Discutir lo que es correcto y lo que es incorrecto sobre la declaración

\ [\ mathbf {x}\ veces\ mathbf {y} =\ nombreoperador {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\]

Ejercicio$$\PageIndex{14}$$

Mostrar que el conjunto de todos los puntos$$\mathbf{x}=(x, y, z)$$ en los$$\mathbb{R}^{3}$$ que satisfacer la ecuación

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

es un plano que pasa por los puntos (0, 0, 0), (1, 2, −1) y (3, 1, 2).

Contestar

Este es el conjunto de todos los puntos que satisfacen$$x-y-z=0$$.

Ejercicio$$\PageIndex{15}$$

Verificar directamente que si$$L: \mathbb{R}^{m} \rightarrow \mathbb{R}^{p}$$ y$$K: \mathbb{R}^{p} \rightarrow \mathbb{R}^{n}$$ son funciones lineales, entonces también$$K \circ L: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$$ es una función lineal.

This page titled 1.6.E: Operaciones con Matrices (Ejercicios) is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform.