Question

PUC–SP–Adaptada)
São dadas as matrizes A = (aij) e B = (bij), C = (cij) quadradas de ordem 3 x 3,
com a matriz A = 2i + 2j;  B = – 2i – 2j
e C = i – j. Determine a matriz:


a)     
D = C – A


b)     
E = B – C


c)     
F = A + B + C


d)     
G = A – B – C


e)     
H = B – A

Answer 1

Resposta:

As matrizes pedidas são:

a)

$D=\begin{pmatrix}-4&-7&-10\\-5&-8&-11\\-6&-9&-12 \end{pmatrix}$

b)

$E=\begin{pmatrix}-4&-5&-6\\-7&-8&-9\\-10&-11&-12 \end{pmatrix}$

c)

$F=\begin{pmatrix}0&-1&-2\\1&0&-1\\2&1&0 \end{pmatrix}$

d)

$G=\begin{pmatrix}8&13&18\\11&16&21\\14&19&24 \end{pmatrix}$

e)

$H=\begin{pmatrix}-8&-12&-16\\-12&-16&-20\\-16&-20&-24 \end{pmatrix}$

Explicação passo a passo:

Para responder a esta questão podemos utilizar a propriedade da soma de matrizes onde podemos somar as definições e por fim obter a matriz pedida.

Sejam as definições das matrizes A, B e C, todas de ordem 3.

$a_{ij}=2i+2j\\\\b_{ij}=-2i-2j\\\\c_{ij}=i-j$

a) D = C - A

$d_{ij}=c_{ij}-a_{ij}\\\\d_{ij}=i-j-(2i+2j)\\\\d_{ij}=-i-3j\\\\D=\begin{pmatrix}d_{11}&d_{12}&d_{13}\\d_{21}&d_{22}&d_{23}\\d_{31}&d_{32}&d_{33} \end{pmatrix}\\\\D=\begin{pmatrix}-4&-7&-10\\-5&-8&-11\\-6&-9&-12 \end{pmatrix}$

b) E = B - C

$e_{ij}=b_{ij}-c_{ij}\\\\e_{ij}=-2i-2j-(i-j)\\\\e_{ij}=-3i-j\\\\E=\begin{pmatrix}e_{11}&e_{12}&e_{13}\\e_{21}&e_{22}&e_{23}\\e_{31}&e_{32}&e_{33} \end{pmatrix}\\\\E=\begin{pmatrix}-4&-5&-6\\-7&-8&-9\\-10&-11&-12 \end{pmatrix}$

c) F = A + B + C

$f_{ij}=a_{ij}+b_{ij}+c_{ij}\\\\f_{ij}=2i+2j-2i-2j+i-j\\\\f_{ij}=i-j\\\\F=\begin{pmatrix}f_{11}&f_{12}&f_{13}\\f_{21}&f_{22}&f_{23}\\f_{31}&f_{32}&f_{33} \end{pmatrix}\\\\F=\begin{pmatrix}0&-1&-2\\1&0&-1\\2&1&0 \end{pmatrix}$

d) G = A - B - C

$g_{ij}=a_{ij}-b_{ij}-c_{ij}\\\\g_{ij}=2i+2j+2i+2j-i+j\\\\g_{ij}=3i+5j\\\\G=\begin{pmatrix}g_{11}&g_{12}&g_{13}\\g_{21}&g_{22}&g_{23}\\g_{31}&g_{32}&g_{33} \end{pmatrix}\\\\G=\begin{pmatrix}8&13&18\\11&16&21\\14&19&24 \end{pmatrix}$

e) H = B - A

$h_{ij}=b_{ij}-a_{ij}\\\\h_{ij}=-2i-2j-(2i+2j)\\\\h_{ij}=-4i-4j\\\\H=\begin{pmatrix}h_{11}&h_{12}&h_{13}\\h_{21}&h_{22}&h_{23}\\h_{31}&h_{32}&h_{33} \end{pmatrix}\\\\H=\begin{pmatrix}-8&-12&-16\\-12&-16&-20\\-16&-20&-24 \end{pmatrix}$