{"id":221238,"date":"2025-02-04T14:10:05","date_gmt":"2025-02-04T17:10:05","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=221238"},"modified":"2025-02-04T14:10:05","modified_gmt":"2025-02-04T17:10:05","slug":"sala-3-problemas","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/sala-3-problemas\/","title":{"rendered":"Sala 3 – Problemas"},"content":{"rendered":"
\n
Determinantes<\/div>\n
Problemas<\/div>\n
\n

\n

Problema 1<\/strong> <\/p>\n

(FEI, 1996) para que o determinante da matriz:
\n[tex]\\qquad \\begin{bmatrix}
\n1+a & -1 \\\\
\n3 & 1-a \\\\
\n\\end{bmatrix}[\/tex]
\nseja nulo, o valor de [tex]a[\/tex], deve ser
\n[tex]\\qquad[\/tex]a) [tex]2[\/tex] ou [tex]-2[\/tex]
\n[tex]\\qquad[\/tex]b) [tex]1[\/tex] ou [tex]3[\/tex]
\n[tex]\\qquad[\/tex]c) [tex]-3[\/tex] ou [tex]5[\/tex]
\n[tex]\\qquad[\/tex]d) [tex]-5[\/tex] ou [tex]3[\/tex]
\n[tex]\\qquad[\/tex]e) [tex]4[\/tex] ou [tex]-4[\/tex]<\/p>\n

\n

\n <\/p>\n
\n

\n

Devemos ter <\/p>\n

[tex]\\qquad \\begin{vmatrix}
\n1+a & -1 \\\\
\n3 & 1-a \\\\
\n\\end{vmatrix}=0[\/tex]
\n[tex]\\qquad (1+a)(1-a) +3=0[\/tex]
\n[tex]\\qquad 1-a^2 +3=0[\/tex]
\n[tex]\\qquad a^2 =4[\/tex]
\n[tex]\\qquad \\boxed{a =\\pm 2}.[\/tex]
\nAlternativa (a)<\/strong>\n<\/div>\n<\/div>\n<\/div>\n

\n

Problema 2<\/strong><\/p>\n

(UniSC, 2017) Dadas as matrizes
\n [tex]\\qquad A =\\begin{bmatrix}
\n1 & 2 \\\\
\n3 & 4 \\\\
\n\\end{bmatrix}[\/tex] e [tex]B = \\begin{bmatrix}
\n-1 & 2 \\\\
\n1 & 0 \\\\
\n\\end{bmatrix},[\/tex]
\no determinante da matriz [tex]A\\cdot B[\/tex] \u00e9
\n[tex]\\qquad[\/tex]a) [tex]4[\/tex]
\n[tex]\\qquad[\/tex]b) [tex]6[\/tex]
\n[tex]\\qquad[\/tex]c) [tex]8[\/tex]
\n[tex]\\qquad[\/tex]d) [tex]12[\/tex]
\n[tex]\\qquad[\/tex]e) [tex]27[\/tex]<\/p>\n

\n

\n <\/p>\n
\n

\n

Pelo teorema de Binet,<\/p>\n

[tex]\\qquad \\det (A\\cdot B) = (\\det A)\\cdot (\\det B)= -2\\cdot (-2) = 4.[\/tex]<\/p>\n

Alternativa (a)<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n

Problema 3<\/strong> <\/p>\n

(UFOP) Considere a matriz [tex]S =\\begin{bmatrix}
\nS_{11} & S_{12} & S_{13} \\\\
\nS_{21} & S_{22} & S_{23} \\\\
\nS_{31} & S_{32} & S_{33} \\\\
\n\\end{bmatrix}[\/tex] dada por [tex]S_{ij} = \\begin{cases}0, \\;\\text{se}\\; i\\lt j\\\\
\ni+j, \\;\\text{se}\\; i=j\\\\
\ni-j, \\;\\text{se}\\; i\\gt j
\n\\end{cases}[\/tex].<\/p>\n

Ent\u00e3o, resolva a inequa\u00e7\u00e3o [tex]\\det S \\gt 3x^2[\/tex].<\/p>\n

\n

\n <\/p>\n
\n

\n

Vamos determinar todos os elementos da matriz [tex]S[\/tex]:<\/p>\n