<h4>Zadaci - Matrice</h4> <p>(a) Neka su \( A = \begin{bmatrix} §§V1(2,4,1)§§ & §§V2(3,5,1)§§ \\ §§V3(1,3,1)§§ & §§V4(4,6,1)§§ \end{bmatrix} \) i \( B = \begin{bmatrix} §§V5(5,7,1)§§ & §§V6(-1,1,0.5)§§ \\ §§V7(2,4,1)§§ & §§V8(0,2,0.5)§§ \end{bmatrix} \). Izračunajte \( A + B \) i \( A - B \).</p> <p>(b) Zadane su matrice \( C = \begin{bmatrix} §§V9(3,5,1)§§ & §§V10(2,4,1)§§ \\ §§V11(-1,1,0.5)§§ & §§V12(6,8,1)§§ \end{bmatrix} \) i \( D = \begin{bmatrix} §§V13(4,6,1)§§ & §§V14(1,3,1)§§ \\ §§V15(2,4,1)§§ & §§V16(-3,-1,0.5)§§ \end{bmatrix} \). Pomnožite matrice \( C \) i \( D \) i zatim izračunajte inverz matrice \( C^{-1} \).</p> <p>(c) Razmotrite matrice \( E = \begin{bmatrix} §§V17(1,3,1)§§ & §§V18(2,4,1)§§ \\ §§V19(3,5,1)§§ & §§V20(4,6,1)§§ \end{bmatrix} \) i \( F = \begin{bmatrix} §§V21(0,2,0.5)§§ & §§V22(5,7,1)§§ \\ §§V23(-2,0,0.5)§§ & §§V24(1,3,1)§§ \end{bmatrix} \). Napišite matricu \( EF \) i izračunajte \( \det(EF) \).</p> <p>(d) Neka je \( G = \begin{bmatrix} §§V25(2,4,1)§§ & §§V26(-1,1,0.5)§§ \\ §§V27(3,5,1)§§ & §§V28(0,2,0.5)§§ \end{bmatrix} \). Pronađite vlastite vrijednosti i odgovarajuće vlastite vektore za matricu \( G \).</p> <p>(e) Zadane su matrice \( H = \begin{bmatrix} §§V29(2,4,1)§§ & §§V30(4,6,1)§§ \\ §§V31(-1,1,0.5)§§ & §§V32(3,5,1)§§ \end{bmatrix} \) i \( I = \begin{bmatrix} §§V33(1,3,1)§§ & §§V34(-2,0,0.5)§§ \\ §§V35(3,5,1)§§ & §§V36(0,2,0.5)§§ \end{bmatrix} \). Provjerite jesu li matrice \( H \) i \( I \) jednake.</p> <p>(f) Funkcija \(p(x)\) definirana je kao \(p(x) = \begin{bmatrix} §§V37(1,3,1)§§ \\ §§V38(2,4,1)§§ \end{bmatrix} \). Izračunajte \( J^K \) (J na potenciju K).</p> <p>(g) Uzmimo matrice \( L = \begin{bmatrix} §§V39(2,4,1)§§ & §§V40(1,3,1)§§ \\ §§V41(-3,-1,0.5)§§ & §§V42(0,2,0.5)§§ \end{bmatrix} \) i \( M = \begin{bmatrix} §§V43(3,5,1)§§ \\ §§V44(2,4,1)§§ \end{bmatrix} \). Pomnožite matricu \( L \) i vektor \( M \) te odredite rezultirajući vektor.</p> <p>(h) Matrica \( N = \begin{bmatrix} §§V45(1,3,1)§§ & §§V46(2,4,1)§§ & §§V47(3,5,1)§§ \\ §§V48(4,6,1)§§ & §§V49(5,7,1)§§ & §§V50(6,8,1)§§ \\ §§V51(7,9,1)§§ & §§V52(8,10,1)§§ & §§V53(9,11,1)§§ \end{bmatrix} \) je kvadratna matrica reda 3. Izračunajte determinantu matrice \( N \).</p> <p>(i) Neka je \( P = \begin{bmatrix} §§V54(2,4,1)§§ & §§V55(-1,1,0.5)§§ \\ §§V56(1,3,1)§§ & §§V57(3,5,1)§§ \end{bmatrix} \). Pronađite inverz matrice \( P^{-1} \).</p> <p>(j) Razmotrite matricu \( Q = \begin{bmatrix} §§V58(4,6,1)§§ & §§V59(1,3,1)§§ \\ §§V60(-2,0,0.5)§§ & §§V61(3,5,1)§§ \end{bmatrix} \). Odredite vlastite vrijednosti i odgovarajuće vlastite vektore za matricu \( Q \).</p>