2.2.1: Ejercicios 2.2

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En Ejercicios\(\PageIndex{1}\) -\(\PageIndex{12}\), los vectores de fila y columna\(\vec{u}\) y\(\vec{v}\) se definen. Encuentre el producto\(\vec{u}\vec{v}\), donde sea posible.

Ejercicio\(\PageIndex{1}\)

\(\vec{u}=\left[\begin{array}{cc}{1}&{-4}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{-2}\\{5}\end{array}\right]\)

Contestar: \(-22\)

Ejercicio\(\PageIndex{2}\)

\(\vec{u}=\left[\begin{array}{cc}{2}&{3}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{7}\\{-4}\end{array}\right]\)

Contestar: \(2\)

Ejercicio\(\PageIndex{3}\)

\(\vec{u}=\left[\begin{array}{cc}{1}&{-1}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{3}\\{3}\end{array}\right]\)

Contestar: \(0\)

Ejercicio\(\PageIndex{4}\)

\(\vec{u}=\left[\begin{array}{cc}{0.6}&{0.8}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{0.6}\\{0.8}\end{array}\right]\)

Contestar: \(1\)

Ejercicio\(\PageIndex{5}\)

\(\vec{u}=\left[\begin{array}{ccc}{1}&{2}&{-1}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{2}\\{1}\\{-1}\end{array}\right]\)

Contestar: \(5\)

Ejercicio\(\PageIndex{6}\)

\(\vec{u}=\left[\begin{array}{ccc}{3}&{2}&{-2}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{-1}\\{0}\\{9}\end{array}\right]\)

Contestar: \(-21\)

Ejercicio\(\PageIndex{7}\)

\(\vec{u}=\left[\begin{array}{ccc}{8}&{-4}&{3}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{2}\\{4}\\{5}\end{array}\right]\)

Contestar: \(15\)

Ejercicio\(\PageIndex{8}\)

\(\vec{u}=\left[\begin{array}{ccc}{-3}&{6}&{1}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{1}\\{-1}\\{1}\end{array}\right]\)

Contestar: \(-8\)

Ejercicio\(\PageIndex{9}\)

\(\vec{u}=\left[\begin{array}{cccc}{1}&{2}&{3}&{4}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{1}\\{-1}\\{1}\\{-1}\end{array}\right]\)

Contestar: \(-2\)

Ejercicio\(\PageIndex{10}\)

\(\vec{u}=\left[\begin{array}{cccc}{6}&{2}&{-1}&{2}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{3}\\{2}\\{9}\\{5}\end{array}\right]\)

Contestar: \(23\)

Ejercicio\(\PageIndex{11}\)

\(\vec{u}=\left[\begin{array}{ccc}{1}&{2}&{3}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{3}\\{2}\end{array}\right]\)

Contestar: No es posible.

Ejercicio\(\PageIndex{12}\)

\(\vec{u}=\left[\begin{array}{cc}{2}&{-5}\end{array}\right]\quad\vec{v}=\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]\)

Contestar: No es posible.

En Ejercicios\(\PageIndex{13}\) -\(\PageIndex{27}\),\(B\) se definen matrices\(A\) y.

Dar las dimensiones de\(A\) y\(B\). Si las dimensiones coinciden correctamente, dar las dimensiones de\(AB\) y\(BA\).
Encuentra los productos\(AB\) y\(BA\), si es posible.

Ejercicio\(\PageIndex{13}\)

\(A=\left[\begin{array}{cc}{1}&{2}\\{-1}&{4}\end{array}\right]\)\(B=\left[\begin{array}{cc}{2}&{5}\\{3}&{-1}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{cc}{8}&{3}\\{10}&{-9}\end{array}\right]\)

\(BA=\left[\begin{array}{cc}{-3}&{24}\\{4}&{2}\end{array}\right]\)

Ejercicio\(\PageIndex{14}\)

\(A=\left[\begin{array}{cc}{3}&{7}\\{2}&{5}\end{array}\right]\)\(B=\left[\begin{array}{cc}{1}&{-1}\\{3}&{-3}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{cc}{24}&{-24}\\{17}&{-17}\end{array}\right]\)

\(BA=\left[\begin{array}{cc}{1}&{2}\\{3}&{6}\end{array}\right]\)

Ejercicio\(\PageIndex{15}\)

\(A=\left[\begin{array}{cc}{3}&{-1}\\{2}&{2}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{1}&{0}&{7}\\{4}&{2}&{9}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{-1}&{-2}&{12}\\{10}&{4}&{32}\end{array}\right]\)

\(BA\)no es posible.

Ejercicio\(\PageIndex{16}\)

\(A=\left[\begin{array}{cc}{0}&{1}\\{1}&{-1}\\{-2}&{-4}\end{array}\right]\)\(B=\left[\begin{array}{cc}{-2}&{0}\\{3}&{8}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{cc}{3}&{8}\\{-5}&{-8}\\{-8}&{-32}\end{array}\right]\)

\(BA\)no es posible.

Ejercicio\(\PageIndex{17}\)

\(A=\left[\begin{array}{ccc}{9}&{4}&{3}\\{9}&{-5}&{9}\end{array}\right]\)\(B=\left[\begin{array}{cc}{-2}&{5}\\{-2}&{-1}\end{array}\right]\)

Contestar

\(AB\)no es posible.

\(BA=\left[\begin{array}{ccc}{27}&{-33}&{39}\\{-27}&{-3}&{-15}\end{array}\right]\)

Ejercicio\(\PageIndex{18}\)

\(A=\left[\begin{array}{cc}{-2}&{-1}\\{9}&{-5}\\{3}&{-1}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{-5}&{6}&{-4}\\{0}&{6}&{-3}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{10}&{-18}&{11}\\{-45}&{24}&{-21}\\{-15}&{12}&{-9}\end{array}\right]\)

\(BA=\left[\begin{array}{cc}{52}&{-21}\\{45}&{-27}\end{array}\right]\)

Ejercicio\(\PageIndex{19}\)

\(A=\left[\begin{array}{cc}{2}&{6}\\{6}&{2}\\{5}&{-1}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{-4}&{5}&{0}\\{-4}&{4}&{-4}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{-32}&{34}&{-24}\\{-32}&{38}&{-8}\\{-16}&{21}&{4}\end{array}\right]\)

\(BA=\left[\begin{array}{cc}{22}&{-14}\\{-4}&{-12}\end{array}\right]\)

Ejercicio\(\PageIndex{20}\)

\(A=\left[\begin{array}{cc}{-5}&{2}\\{-5}&{-2}\\{-5}&{-4}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{0}&{-5}&{6}\\{-5}&{-3}&{-1}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{-10}&{19}&{-32}\\{10}&{31}&{-28}\\{20}&{37}&{-26}\end{array}\right]\)

\(BA=\left[\begin{array}{cc}{-5}&{-14}\\{45}&{0}\end{array}\right]\)

Ejercicio\(\PageIndex{21}\)

\(A=\left[\begin{array}{cc}{8}&{-2}\\{4}&{5}\\{2}&{-5}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{-5}&{1}&{-5}\\{8}&{3}&{-2}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{-56}&{2}&{-36}\\{20}&{19}&{-30}\\{-50}&{-13}&{0}\end{array}\right]\)

\(BA=\left[\begin{array}{cc}{-46}&{40}\\{72}&{9}\end{array}\right]\)

Ejercicio\(\PageIndex{22}\)

\(A=\left[\begin{array}{cc}{1}&{4}\\{7}&{6}\end{array}\right]\)\(B=\left[\begin{array}{cccc}{1}&{-1}&{-5}&{5}\\{-2}&{1}&{3}&{-5}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{cccc}{-7}&{3}&{7}&{-15}\\{-5}&{-1}&{-17}&{5}\end{array}\right]\)

\(BA\)no es posible.

Ejercicio\(\PageIndex{23}\)

\(A=\left[\begin{array}{cc}{-1}&{5}\\{6}&{7}\end{array}\right]\)\(B=\left[\begin{array}{cccc}{5}&{-3}&{-4}&{-4}\\{-2}&{-5}&{-5}&{-1}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{cccc}{-15}&{-22}&{-21}&{-1}\\{16}&{-53}&{-59}&{-31}\end{array}\right]\)

\(BA\)no es posible.

Ejercicio\(\PageIndex{24}\)

\(A=\left[\begin{array}{ccc}{-1}&{2}&{1}\\{-1}&{2}&{-1}\\{0}&{0}&{-2}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{0}&{0}&{-2}\\{1}&{2}&{-1}\\{1}&{0}&{0}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{3}&{4}&{0}\\{1}&{4}&{0}\\{-2}&{0}&{0}\end{array}\right]\)

\(BA=\left[\begin{array}{ccc}{0}&{0}&{4}\\{-3}&{6}&{1}\\{-1}&{2}&{1}\end{array}\right]\)

Ejercicio\(\PageIndex{25}\)

\(A=\left[\begin{array}{ccc}{-1}&{1}&{1}\\{-1}&{-1}&{-2}\\{1}&{1}&{-2}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{-2}&{-2}&{-2}\\{0}&{-2}&{0}\\{-2}&{0}&{2}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{0}&{0}&{4}\\{6}&{4}&{-2}\\{2}&{-4}&{-6}\end{array}\right]\)

\(BA=\left[\begin{array}{ccc}{2}&{-2}&{6}\\{2}&{2}&{4}\\{4}&{0}&{-6}\end{array}\right]\)

Ejercicio\(\PageIndex{26}\)

\(A=\left[\begin{array}{ccc}{-4}&{3}&{3}\\{-5}&{-1}&{-5}\\{-5}&{0}&{-1}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{0}&{5}&{0}\\{-5}&{-4}&{3}\\{5}&{-4}&{3}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{0}&{-44}&{18}\\{-20}&{-1}&{-18}\\{-5}&{-21}&{-3}\end{array}\right]\)

\(BA=\left[\begin{array}{ccc}{-25}&{-5}&{-25}\\{25}&{-11}&{2}\\{-15}&{19}&{32}\end{array}\right]\)

Ejercicio\(\PageIndex{27}\)

\(A=\left[\begin{array}{ccc}{-4}&{-1}&{3}\\{2}&{-3}&{5}\\{1}&{5}&{3}\end{array}\right]\)\(B=\left[\begin{array}{ccc}{-2}&{4}&{3}\\{-1}&{1}&{-1}\\{4}&{0}&{2}\end{array}\right]\)

Contestar

\(AB=\left[\begin{array}{ccc}{21}&{-17}&{-5}\\{19}&{5}&{19}\\{5}&{9}&{4}\end{array}\right]\)

\(BA=\left[\begin{array}{ccc}{19}&{5}&{23}\\{5}&{-7}&{-1}\\{-14}&{6}&{18}\end{array}\right]\)

En Ejercicios\(\PageIndex{28}\) -\(\PageIndex{33}\), se dan una matriz diagonal\(D\) y una matriz\(A\). Encuentre los productos\(DA\) y\(AD\), cuando sea posible.

Ejercicio\(\PageIndex{28}\)

\(D=\left[\begin{array}{cc}{3}&{0}\\{0}&{-1}\end{array}\right]\)\(A=\left[\begin{array}{cc}{2}&{4}\\{6}&{8}\end{array}\right]\)

Contestar

\(DA=\left[\begin{array}{cc}{6}&{-4}\\{18}&{-8}\end{array}\right]\)

\(AD=\left[\begin{array}{cc}{6}&{12}\\{-6}&{-8}\end{array}\right]\)

Ejercicio\(\PageIndex{29}\)

\(D=\left[\begin{array}{cc}{4}&{0}\\{0}&{-3}\end{array}\right]\)\(A=\left[\begin{array}{cc}{1}&{2}\\{1}&{2}\end{array}\right]\)

Contestar

\(DA=\left[\begin{array}{cc}{4}&{-6}\\{4}&{-6}\end{array}\right]\)

\(AD=\left[\begin{array}{cc}{4}&{8}\\{-3}&{-6}\end{array}\right]\)

Ejercicio\(\PageIndex{30}\)

\(D=\left[\begin{array}{ccc}{-1}&{0}&{0}\\{0}&{2}&{0}\\{0}&{0}&{3}\end{array}\right]\)\(A=\left[\begin{array}{ccc}{1}&{2}&{3}\\{4}&{5}&{6}\\{7}&{8}&{9}\end{array}\right]\)

Contestar

\(DA=\left[\begin{array}{ccc}{-1}&{4}&{9}\\{-4}&{10}&{18}\\{-7}&{16}&{27}\end{array}\right]\)

\(AD=\left[\begin{array}{ccc}{-1}&{-2}&{-3}\\{8}&{10}&{12}\\{21}&{24}&{27}\end{array}\right]\)

Ejercicio\(\PageIndex{31}\)

\(D=\left[\begin{array}{ccc}{1}&{1}&{1}\\{2}&{2}&{2}\\{-3}&{-3}&{-3}\end{array}\right]\)\(A=\left[\begin{array}{ccc}{2}&{0}&{0}\\{0}&{-3}&{0}\\{0}&{0}&{5}\end{array}\right]\)

Contestar

\(DA=\left[\begin{array}{ccc}{2}&{2}&{2}\\{-6}&{-6}&{-6}\\{-15}&{-15}&{-15}\end{array}\right]\)

\(AD=\left[\begin{array}{ccc}{2}&{-3}&{5}\\{4}&{-6}&{10}\\{-6}&{9}&{-15}\end{array}\right]\)

Ejercicio\(\PageIndex{32}\)

\(D=\left[\begin{array}{cc}{d_{1}}&{0}\\{0}&{d_{2}}\end{array}\right]\)\(A=\left[\begin{array}{cc}{a}&{b}\\{c}&{d}\end{array}\right]\)

Contestar

\(DA=\left[\begin{array}{cc}{d_{1}a}&{d_{1}b}\\{d_{2}c}&{d_{2}d}\end{array}\right]\)

\(AD=\left[\begin{array}{cc}{d_{1}a}&{d_{2}b}\\{d_{1}c}&{d_{2}d}\end{array}\right]\)

Ejercicio\(\PageIndex{33}\)

\(D=\left[\begin{array}{ccc}{d_{1}}&{0}&{0}\\{0}&{d_{2}}&{0}\\{0}&{0}&{d_{3}}\end{array}\right]\)\(A=\left[\begin{array}{ccc}{a}&{b}&{c}\\{d}&{e}&{f}\\{g}&{h}&{i}\end{array}\right]\)

Contestar

\(DA=\left[\begin{array}{ccc}{d_{1}a}&{d_{1}b}&{d_{1}c}\\{d_{2}d}&{d_{2}e}&{d_{2}f}\\{d_{3}g}&{d_{3}h}&{d_{3}i}\end{array}\right]\)

\(AD=\left[\begin{array}{ccc}{d_{1}a}&{d_{2}b}&{d_{3}c}\\{d_{1}d}&{d_{2}e}&{d_{3}f}\\{d_{1}g}&{d_{2}h}&{d_{3}i}\end{array}\right]\)

En Ejercicios\(\PageIndex{34}\) -\(\PageIndex{39}\), se dan una matriz\(A\) y un vector\(\vec{x}\). Encuentra el producto\(A\vec{x}\).

Ejercicio\(\PageIndex{34}\)

\(A=\left[\begin{array}{cc}{2}&{3}\\{1}&{-1}\end{array}\right]\),\(\vec{x}=\left[\begin{array}{c}{4}\\{9}\end{array}\right]\)

Contestar: \(A\vec{x}=\left[\begin{array}{c}{35}\\{-5}\end{array}\right]\)

Ejercicio\(\PageIndex{35}\)

\(A=\left[\begin{array}{cc}{-1}&{4}\\{7}&{3}\end{array}\right]\),\(\vec{x}=\left[\begin{array}{c}{2}\\{-1}\end{array}\right]\)

Contestar: \(A\vec{x}=\left[\begin{array}{c}{-6}\\{11}\end{array}\right]\)

Ejercicio\(\PageIndex{36}\)

\(A=\left[\begin{array}{ccc}{2}&{0}&{3}\\{1}&{1}&{1}\\{3}&{-1}&{2}\end{array}\right]\),\(\vec{x}=\left[\begin{array}{c}{1}\\{4}\\{2}\end{array}\right]\)

Contestar: \(A\vec{x}=\left[\begin{array}{c}{8}\\{7}\\{3}\end{array}\right]\)

Ejercicio\(\PageIndex{37}\)

\(A=\left[\begin{array}{ccc}{-2}&{0}&{3}\\{1}&{1}&{-2}\\{4}&{2}&{-1}\end{array}\right]\),\(\vec{x}=\left[\begin{array}{c}{4}\\{3}\\{1}\end{array}\right]\)

Contestar: \(A\vec{x}=\left[\begin{array}{c}{-5}\\{5}\\{21}\end{array}\right]\)

Ejercicio\(\PageIndex{38}\)

\(A=\left[\begin{array}{cc}{2}&{-1}\\{4}&{3}\end{array}\right]\),\(\vec{x}=\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\)

Contestar: \(A\vec{x}=\left[\begin{array}{c}{2x_{1}-x_{2}}\\{4x_{1}+3x_{2}}\end{array}\right]\)

Ejercicio\(\PageIndex{39}\)

\(A=\left[\begin{array}{ccc}{1}&{2}&{3}\\{1}&{0}&{2}\\{2}&{3}&{1}\end{array}\right]\),\(\vec{x}=\left[\begin{array}{c}{x_{1}}\\{x_{2}}\\{x_{3}}\end{array}\right]\)

Contestar: \(A\vec{x}\left[\begin{array}{c}{x_{1}+2x_{2}+3x_{3}}\\{x_{1}+2x_{3}}\\{2x_{1}+3x_{2}+x_{3}}\end{array}\right]\)

Ejercicio\(\PageIndex{40}\)

Vamos\(A=\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right]\). Encontrar\(A^{2}\) y\(A^{3}\).

Contestar: \(A^{2}=\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\);\(A^{3}=\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right]\)

Ejercicio\(\PageIndex{41}\)

Vamos\(A=\left[\begin{array}{cc}{2}&{0}\\{0}&{3}\end{array}\right]\). Encontrar\(A^{2}\) y\(A^{3}\).

Contestar: \(A^{2}=\left[\begin{array}{cc}{4}&{0}\\{0}&{9}\end{array}\right]\);\(A^{3}=\left[\begin{array}{cc}{8}&{0}\\{0}&{27}\end{array}\right]\)

Ejercicio\(\PageIndex{42}\)

Vamos\(A=\left[\begin{array}{ccc}{-1}&{0}&{0}\\{0}&{3}&{0}\\{0}&{0}&{5}\end{array}\right]\). Encontrar\(A^{2}\) y\(A^{3}\).

Contestar: \(A^{2}=\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{9}&{0}\\{0}&{0}&{25}\end{array}\right]\);\(A^{3}=\left[\begin{array}{ccc}{-1}&{0}&{0}\\{0}&{27}&{0}\\{0}&{0}&{125}\end{array}\right]\)

Ejercicio\(\PageIndex{43}\)

Vamos\(A=\left[\begin{array}{ccc}{0}&{1}&{0}\\{0}&{0}&{1}\\{1}&{0}&{0}\end{array}\right]\). Encontrar\(A^{2}\) y\(A^{3}\).

Contestar: \(A^{2}=\left[\begin{array}{ccc}{0}&{0}&{1}\\{1}&{0}&{0}\\{0}&{1}&{0}\end{array}\right]\);\(A^{3}=\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\end{array}\right]\)

Ejercicio\(\PageIndex{44}\)

Vamos\(A=\left[\begin{array}{ccc}{0}&{0}&{1}\\{0}&{0}&{0}\\{0}&{1}&{0}\end{array}\right]\). Encontrar\(A^{2}\) y\(A^{3}\).

Contestar: \(A^{2}=\left[\begin{array}{ccc}{0}&{1}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]\);\(A^{3}=\left[\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]\)

Ejercicio\(\PageIndex{45}\)

En el texto, lo afirmamos\((A+B)^{2}\neq A^{2}+2AB+B^{2}\). Investigamos esa afirmación aquí.

Dejar\(A=\left[\begin{array}{cc}{5}&{3}\\{-3}&{-2}\end{array}\right]\) y dejar\(B=\left[\begin{array}{cc}{-5}&{-5}\\{-2}&{1}\end{array}\right]\). Cómponlo\(A+B\).
Encuentra\((A+B)^{2}\) usando tu respuesta de (a).
Cómponlo\(A^{2}+2AB+B^{2}\).
¿Los resultados de (a) y (b) son los mismos?
Amplíe cuidadosamente la expresión\((A+B)^{2}=(A+B)(A+B)\) y muestre por qué esto no es igual a\(A^{2}+2AB+B^{2}\).

Contestar

\(\left[\begin{array}{cc}{0}&{-2}\\{-5}&{-1}\end{array}\right]\)
\(\left[\begin{array}{cc}{10}&{2}\\{5}&{11}\end{array}\right]\)
\(\left[\begin{array}{cc}{-11}&{-15}\\{37}&{32}\end{array}\right]\)
No
\((A+B)(A+B)=AA+AB+BA+BB=A^{2}+AB+BA+B^{2}\)