OPERACIONES+CON+MATRICES

**//__OPERACIONES CON MATRICES__//**

==Dadas dos matrices de la misma dimensión, **A=(aij)** y **B=(bij)**, se define la matriz suma como: **A+B=(aij+bij)**. Es decir, aquella matriz cuyos elementos se obtienen: sumando los elementos de las dos matrices que ocupan la misma misma posición. == 

Propiedades de la suma de matrices
**Interna:** **La suma de dos matrices de orden m x n es otra matriz dimensión m x n.** **Asociativa:** **A + (B + C) = (A + B) + C** **Elemento neutro:** **A + 0 = A** Donde **O** es la matriz nula de la misma dimensión que la matriz A. Elemento opuesto: A + (-A) = O La matriz opuesta es aquella en que todos los elementos están cambiados de signo. Conmutativa: A + B = B + A

<span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Producto de matrices
<span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">**Mm x n x Mn x p = M m x p**

<span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Matriz inversa
<span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">**A · A-1 = A-1 · A = I** <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">**(A · B)-1 = B-1 · A-1** <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">**(A-1)-1 = A** <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">**(k · A)-1 = k-1 · A-1**

<span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Cálculo de la matriz inversa
<span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Ejercicios <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Dadas las matrices: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Calcular: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">**A + B; A - B; A x B; B x A; At**. <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">

<span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">

<span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span class="actividades" style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Sean las matrices: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Efectuar las siguientes operaciones: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">**(A + B) 2; (A - B) 2; (B) 3; A · B t · C.** <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Dadas las matrices: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"><span class="numero_v">1 Justificar si son posibles los siguientes productos: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"><span class="numero_r">1 **(A t · B ) · C** <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">(At3 x 2 · B2 x 2 ) · C3 x 2 = (At · B )3 x 2 · C3 x 2 <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">No se puede efectuar el producto porque el número de columnas de (At · B ) no coincide con el nº de filas de C. <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"><span class="numero_r">2 **(B · Ct ) · At** <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">(B2 x 2 · Ct2 x 3 ) · At3 x 2 = (B · C )2 x 3 · At3 x 2 = <span class="sol" style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">=(B · C t · A t ) 2 x 2 <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"><span class="numero_v">2 Determinar la dimensión de **M** para que pueda efectuarse el producto **A · M · C** <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">A3 x 2 · Mm x n · C3 x 2 m = 2 <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"><span class="numero_v">3 Determina la dimensión de **M** para que **Ct · M** sea una matriz cuadrada. <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Ct2 x 3 · Mm x n m = 3 n = 3 <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Demostrar que: **A2 - A - 2 I = 0**, siendo: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Sea A la matriz <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Por qué matriz hay que premultiplicar la matriz para que resulte la matriz. <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Hallar la matriz inversa de: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Calcular el rango de las siguientes matrices: <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">|2|=2 ≠0 <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span class="sol" style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">r(A) = 2 <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span class="sol" style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">r(B) = 4 <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">Eliminamos la tercera columna por ser nula, la cuarta por ser proporcional a la primera, y la quinta porque combinación lineal de la primera y segunda: c5 = -2 **·** c1 + c2 <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;"> <span class="sol" style="color: #ff00ff; font-family: "Comic Sans MS",cursive; font-size: 120%;">r(C) = 2