{"id":16366,"date":"2026-07-20T05:19:07","date_gmt":"2026-07-20T05:19:07","guid":{"rendered":"https:\/\/www.vedprep.com\/exams\/?p=16366"},"modified":"2026-07-20T05:19:07","modified_gmt":"2026-07-20T05:19:07","slug":"cayley-hamilton-theorem-cuet-pg","status":"publish","type":"post","link":"https:\/\/www.vedprep.com\/exams\/cuet-pg\/cayley-hamilton-theorem-cuet-pg\/","title":{"rendered":"Cayley-hamilton Theorem for Cuet Pg: Ultimate"},"content":{"rendered":"
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Ultimate Cayley-Hamilton Theorem Guide For CUET PG 2024<\/h1>\n

Every CUET PG aspirant must master the Cayley-Hamilton theorem For CUET PG<\/strong>\u2014a cornerstone of linear algebra that guarantees every square matrix satisfies its own characteristic equation. This theorem isn’t just theoretical; it’s a practical tool for solving complex matrix problems, simplifying calculations, and excelling in competitive exams like CUET PG, CSIR NET, and IIT JAM.<\/strong><\/p>\n

In this definitive guide, we’ll break down the Cayley-Hamilton theorem For CUET PG<\/strong> from its mathematical foundations to real-world applications, ensuring you’re fully prepared for your exam. Whether you’re solving for matrix powers or analyzing system stability, this theorem will be your secret weapon.<\/p>\n

Cayley-hamilton Theorem for Cuet Pg: Key Concepts<\/h2>\n

The Cayley-Hamilton theorem For CUET PG<\/strong> states that for any square matrix A<\/em> of size n\u00d7n<\/em>, the characteristic polynomial p(\u03bb) = det(A – \u03bbI)<\/em> satisfies p(A) = 0<\/em>. This means substituting the matrix A<\/em> itself into its characteristic equation yields the zero matrix. For example, if the characteristic equation is \u03bb\u00b2 – 5\u03bb + 6 = 0<\/em>, then A\u00b2 – 5A + 6I = 0<\/em>.<\/p>\n

This theorem is not just theoretical<\/strong>\u2014it’s a game-changer for CUET PG students. The Cayley-Hamilton theorem For CUET PG<\/strong> allows you to reduce higher powers of a matrix to lower ones, making complex calculations manageable. For instance, if you need to compute A\u2075<\/em>, you can express it in terms of A\u00b2<\/em>, A<\/em>, and I<\/em> using the theorem.<\/p>\n

Why Does the Cayley-Hamilton Theorem For CUET PG<\/strong> Work?<\/h3>\n

The proof relies on the fact that the minimal polynomial of a matrix divides its characteristic polynomial. Since the minimal polynomial always satisfies m(A) = 0<\/em>, and the characteristic polynomial is a multiple of the minimal polynomial, the Cayley-Hamilton theorem For CUET PG<\/strong> follows naturally. This connection between minimal and characteristic polynomials is crucial for understanding why the theorem holds.<\/p>\n

Step-by-Step Application of Cayley-Hamilton Theorem For CUET PG<\/strong><\/h2>\n

Let\u2019s apply the Cayley-Hamilton theorem For CUET PG<\/strong> to a practical example. Consider the matrix:<\/p>\n

A = [1  1; 0  2]<\/pre>\n
The characteristic equation is derived as follows:<\/p>\n
    \n
  1. Compute det(A – \u03bbI)<\/em>:<\/li>\n
  2. det([1-\u03bb  1; 0  2-\u03bb]) = (1-\u03bb)(2-\u03bb) = \u03bb\u00b2 – 3\u03bb + 2<\/em><\/li>\n
  3. Set the equation to zero: \u03bb\u00b2 – 3\u03bb + 2 = 0<\/em><\/li>\n
  4. Substitute A<\/em> into the equation: A\u00b2 – 3A + 2I = 0<\/em><\/li>\n<\/ol>\n
    This simplifies to A\u00b2 = 3A – 2I<\/em>, a powerful relationship that can be used to compute higher powers of A<\/em> efficiently. For example, A\u00b3 = A\u00b7A\u00b2 = A(3A – 2I) = 3A\u00b2 – 2A = 3(3A – 2I) – 2A = 7A – 6I<\/em>.<\/p>\n
    This method is essential for CUET PG<\/strong> because it reduces the computational complexity of matrix operations, saving time during exams.<\/p>\n
    Key Concepts of Cayley-Hamilton Theorem For CUET PG<\/strong> You Must Know<\/h2>\n
    The Cayley-Hamilton theorem For CUET PG<\/strong> revolves around two critical ideas:<\/p>\n