3.3.1: Ejercicios 3.3

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En Ejercicios\(\PageIndex{1}\) -\(\PageIndex{8}\), encuentra el determinante de la\(2\times 2\) matriz.

Ejercicio\(\PageIndex{1}\)

\(\left[\begin{array}{cc}{10}&{7}\\{8}&{9}\end{array}\right]\)

Contestar: \(34\)

Ejercicio\(\PageIndex{2}\)

\(\left[\begin{array}{cc}{6}&{-1}\\{-7}&{8}\end{array}\right]\)

Contestar: \(41\)

Ejercicio\(\PageIndex{3}\)

\(\left[\begin{array}{cc}{-1}&{-7}\\{-5}&{9}\end{array}\right]\)

Contestar: \(-44\)

Ejercicio\(\PageIndex{4}\)

\(\left[\begin{array}{cc}{-10}&{-1}\\{-4}&{7}\end{array}\right]\)

Contestar: \(-74\)

Ejercicio\(\PageIndex{5}\)

\(\left[\begin{array}{cc}{8}&{10}\\{2}&{-3}\end{array}\right]\)

Contestar: \(-44\)

Ejercicio\(\PageIndex{6}\)

\(\left[\begin{array}{cc}{10}&{-10}\\{-10}&{0}\end{array}\right]\)

Contestar: \(-100\)

Ejercicio\(\PageIndex{7}\)

\(\left[\begin{array}{cc}{1}&{-3}\\{7}&{7}\end{array}\right]\)

Contestar: \(28\)

Ejercicio\(\PageIndex{8}\)

\(\left[\begin{array}{cc}{-4}&{-5}\\{-1}&{-4}\end{array}\right]\)

Contestar: \(11\)

En Ejercicios\(\PageIndex{9}\) -\(\PageIndex{12}\),\(A\) se da una matriz.

Construir las submatrices utilizadas para computar los menores\(A_{1,1}\),\(A_{1,2}\), y\(A_{1,3}\).
Encuentra los cofactores\(C_{1,1}\),\(C_{1,2}\), y\(C_{1,3}\).

Ejercicio\(\PageIndex{9}\)

\(\left[\begin{array}{ccc}{-7}&{-3}&{10}\\{3}&{7}&{6}\\{1}&{6}&{10}\end{array}\right]\)

Contestar

Las submatrices son\(\left[\begin{array}{cc}{7}&{6}\\{6}&{10}\end{array}\right]\),\(\left[\begin{array}{cc}{3}&{6}\\{1}&{10}\end{array}\right]\), y\(\left[\begin{array}{cc}{3}&{7}\\{1}&{6}\end{array}\right]\), respectivamente.
\(C_{1,2} = 34\),\(C_{1,2} = −24\),\(C_{1,3} = 11\)

Ejercicio\(\PageIndex{10}\)

\(\left[\begin{array}{ccc}{-2}&{-9}&{6}\\{-10}&{-6}&{8}\\{0}&{-3}&{-2}\end{array}\right]\)

Contestar

Las submatrices son\(\left[\begin{array}{cc}{-6}&{8}\\{-3}&{-2}\end{array}\right]\),\(\left[\begin{array}{cc}{-10}&{8}\\{0}&{-2}\end{array}\right]\), y\(\left[\begin{array}{cc}{10}&{-6}\\{0}&{-3}\end{array}\right]\), respectivamente.
\(C_{1,2} = 36\),\(C_{1,2} = −20\),\(C_{1,3} = -30\)

Ejercicio\(\PageIndex{11}\)

\(\left[\begin{array}{ccc}{-5}&{-3}&{3}\\{-3}&{3}&{10}\\{-9}&{3}&{9}\end{array}\right]\)

Contestar

Las submatrices son\(\left[\begin{array}{cc}{3}&{10}\\{3}&{9}\end{array}\right]\),\(\left[\begin{array}{cc}{-3}&{10}\\{-9}&{9}\end{array}\right]\), y\(\left[\begin{array}{cc}{-3}&{3}\\{-9}&{3}\end{array}\right]\), respectivamente.
\(C_{1,2} = -3\),\(C_{1,2} = −63\),\(C_{1,3} = 18\)

Ejercicio\(\PageIndex{12}\)

\(\left[\begin{array}{ccc}{-6}&{-4}&{6}\\{-8}&{0}&{0}\\{-10}&{8}&{-1}\end{array}\right]\)

Contestar

Las submatrices son\(\left[\begin{array}{ccc}{-6}&{-4}&{6}\\{-8}&{0}&{0}\\{-10}&{8}&{-1}\end{array}\right]\),\(\left[\begin{array}{cc}{-8}&{0}\\{-10}&{-1}\end{array}\right]\), y\(\left[\begin{array}{cc}{-8}&{0}\\{-10}&{8}\end{array}\right]\), respectivamente.
\(C_{1,2} = 0\),\(C_{1,2} = −8\),\(C_{1,3} = -64\)

En Ejercicios\(\PageIndex{13}\) —\(\PageIndex{24}\), encuentra el determinante de la matriz dada usando expansión de cofactores a lo largo de la primera fila.

Ejercicio\(\PageIndex{13}\)

\(\left[\begin{array}{ccc}{3}&{2}&{3}\\{-6}&{1}&{-10}\\{-8}&{-9}&{-9}\end{array}\right]\)

Contestar: \(-59\)

Ejercicio\(\PageIndex{14}\)

\(\left[\begin{array}{ccc}{8}&{-9}&{-2}\\{-9}&{9}&{-7}\\{5}&{-1}&{9}\end{array}\right]\)

Contestar: \(250\)

Ejercicio\(\PageIndex{15}\)

\(\left[\begin{array}{ccc}{-4}&{3}&{-4}\\{-4}&{-5}&{3}\\{3}&{-4}&{5}\end{array}\right]\)

Contestar: \(15\)

Ejercicio\(\PageIndex{16}\)

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

Contestar: \(-52\)

Ejercicio\(\PageIndex{17}\)

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

Contestar: \(3\)

Ejercicio\(\PageIndex{18}\)

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

Contestar: \(0\)

Ejercicio\(\PageIndex{19}\)

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

Contestar: \(0\)

Ejercicio\(\PageIndex{20}\)

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

Contestar: \(1\)

Ejercicio\(\PageIndex{21}\)

\(\left[\begin{array}{cccc}{0}&{0}&{-1}&{-1}\\{1}&{1}&{0}&{1}\\{1}&{1}&{-1}&{0}\\{-1}&{0}&{1}&{0}\end{array}\right]\)

Contestar: \(0\)

Ejercicio\(\PageIndex{22}\)

\(\left[\begin{array}{cccc}{-1}&{0}&{0}&{-1}\\{-1}&{0}&{0}&{1}\\{1}&{1}&{1}&{0}\\{1}&{0}&{-1}&{-1}\end{array}\right]\)

Contestar: \(2\)

Ejercicio\(\PageIndex{23}\)

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

Contestar: \(-113\)

Ejercicio\(\PageIndex{24}\)

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

Contestar: \(179\)

Ejercicio\(\PageIndex{25}\)

Dejar\(A\) ser una\(2\times 2\) matriz;

\[A=\left[\begin{array}{cc}{a}&{b}\\{c}&{d}\end{array}\right]. \nonumber \]

Mostrar\(\text{det}(A)=ad-bc\) por qué calculando la expansión del cofactor\(A\) a lo largo de la primera fila.

Contestar: Pista:\(C_{1,1}=d\).