Question

Dadas as matrizes A2x2, com aij = 3i + 4j e B2x2, com bij = 4i – 3j. Calcule C², sendo que C = A + B. OBS: C² = C.C

Answer 1

Explicação passo-a-passo:

As matrizes A₂ₓ₂ e B₂ₓ₂ são matrizes quadradas de ordem 2 (2 linhas e 2 colunas).

A matriz A será

    $A=\left[\begin{array}{ccc}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{array}\right]$

e a B será

    $B=\left[\begin{array}{ccc}b_{11}&b_{12}\\b_{21}&b_{22}\\\end{array}\right]$

Sendo $a_{ij} =3i+4j$, a matriz A ficará

    a₁₁ = 3 × 1 + 4 × 1 = 3 + 4 = 7

    a₁₂ = 3 × 1 + 4 × 2 = 3 + 8 = 11

    a₂₁ = 3 × 2 + 4 × 1 = 6 + 4 = 10

    a₂₂ = 3 × 2 + 4 × 2 = 6 + 8 = 14

e $b_{ij}=4i-3j$, a matriz B ficará

    b₁₁ = 4 × 1 - 3 × 1 = 4 - 3 = 1

    b₁₂ = 4 × 1 - 3 × 2 = 4 - 6 = -2

    b₂₁ = 4 × 2 - 3 × 1 = 8 - 3 = 5

    b₂₂ = 4 × 2 - 3 × 2 = 8 - 6 = 2

Então, as matrizes A e B ficarão

    $A=\left[\begin{array}{ccc}7&11\\10&14\\\end{array}\right]$     e     $B=\left[\begin{array}{ccc}1&-2\\5&2\\\end{array}\right]$

Cálculo de C

    $C=A+B$

    $C=\left[\begin{array}{ccc}7&11\\10&14\\\end{array}\right]+\left[\begin{array}{ccc}1&-2\\5&2\\\end{array}\right]$

    $C=\left[\begin{array}{ccc}7+1&11+(-2)\\10+5&14+2\\\end{array}\right]$

    $C=\left[\begin{array}{ccc}8&9\\15&16\\\end{array}\right]$

Cálculo de C²

    $C^{2}=\left[\begin{array}{ccc}8&9\\15&16\\\end{array}\right].\left[\begin{array}{ccc}8&9\\15&16\\\end{array}\right]$

    $C^{2}=\left[\begin{array}{ccc}8.8+9.15&8.9+9.16\\15.8+16.15&15.9+16.16\\\end{array}\right]$

    $C^{2}=\left[\begin{array}{ccc}64+135&72+144\\120+240&135+256\\\end{array}\right]$

    $C^{2}=\left[\begin{array}{ccc}199&216\\360&391\\\end{array}\right]$