st","extracted_subject":"MATH","extracted_chapter":"DETERMINANTS AND MATRICES","extracted_topic":"SPECIAL TYPES OF MATRICES, TRANSPOSE, ADJOINT AND INVERSE OF MATRICES","questionPreviewUrl":"https://cdn.pureessence.tech/question-previews/packed_canvases/canvas_2206.png?top_left_x=0&top_left_y=1463&width=532&height=215&app=solveeit"},"seo_metadata":{"keywords":[]},"doubt_solver_data":{"concept_summary":[]},"embeddings":[],"is_hidden":false,"_id":"66d23e74425d76b7ab320fa8","slug":"if-a-satisfies-the-equation-xsup3sup-5xsup2sup-4x-","content":"If A satisfies the equation x³ – 5x² + 4x + k = 0, then A^–1 exists if –","options":[{"_id":"68a95d827f15af4716d575da","text":"k¹ 1","sequence":0,"isCorrect":false},{"_id":"68a95d827f15af4716d575db","text":"k ¹ 3","sequence":1,"isCorrect":false},{"_id":"68a95d827f15af4716d575dc","text":"k ¹ – 1","sequence":2,"isCorrect":false},{"_id":"68a95d827f15af4716d575dd","text":"None","sequence":3,"isCorrect":true}],"correct_answer":"None","explanation":"A³ – 5A² + 5A+ k.I = 0\n\nA^–1 (A³ – 5A² + 5A + kI) = A^–1 × (0)\n\nA² – 5A + 5I + kA^–1 = 0\n\nA^–1 = – \\(\\frac{1}{k}\\) (A² – 5A + 5I)\n\n\\\\ A^–1 exists if k ¹ 0","question_type":"single_choice","appeared_in":[],"created_at":"2024-08-30T21:49:40.392Z","__v":0,"vector_store_id":"7f738703-1ecd-4c12-8c14-a0f440fd37eb","updated_at":"2024-08-30T21:49:40.392Z","isStarred":false},"params":{"questionSlug":"if-a-satisfies-the-equation-xsup3sup-5xsup2sup-4x-"}}]}]]

Question: If A satisfies the equation x<sup>3</sup> – 5x<sup>2</sup> + 4x + k = 0, then A<sup>–1</sup> exists ...

Solution