If A is a square matrix of order 3 satisfying A^3-11 A^2+36 A-30 I=0 then A^5 is given by (a)

step 1 So in the given question we are told that the matrix A, the matrix A is the matrix a is a square

If A is a square matrix of order 3 satisfying A^3-11 A^2+36...

If $\mathrm{A}$ is a square matrix of order 3 satisfying $\mathrm{A}^{3}-11 \mathrm{~A}^{2}+36 \mathrm{~A}-30 \mathrm{I}=0$ then $\mathrm{A}^{5}$ is given by (a) $569 \mathrm{~A}^{2}-2730 \mathrm{~A}+2550 \mathrm{I}$ (b) $569 \mathrm{~A}^{2}+2730 \mathrm{~A}+2550 \mathrm{I}$ (c) $569 \mathrm{~A}^{2}-2730 \mathrm{~A}-2550 \mathrm{I}$ (d) $569 \mathrm{~A}^{2}+2730 \mathrm{~A}-2550 \mathrm{I}$

If $\mathrm{A}$ is a square matrix of order 3 satisfying $\mathrm{A}^{3}-11 \mathrm{~A}^{2}+36 \mathrm{~A}-30 \mathrm{I}=0$ then $\mathrm{A}^{5}$ is given by
(a) $569 \mathrm{~A}^{2}-2730 \mathrm{~A}+2550 \mathrm{I}$
(b) $569 \mathrm{~A}^{2}+2730 \mathrm{~A}+2550 \mathrm{I}$
(c) $569 \mathrm{~A}^{2}-2730 \mathrm{~A}-2550 \mathrm{I}$
(d) $569 \mathrm{~A}^{2}+2730 \mathrm{~A}-2550 \mathrm{I}$

Key Concepts

Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial. This principle allows one to replace higher powers of the matrix with a linear combination of lower powers, facilitating the simplification of matrix expressions and the computation of higher power forms.

Characteristic and Minimal Polynomials

The characteristic polynomial of a matrix, or in some cases the minimal polynomial, encapsulates the essential algebraic properties of the matrix. When a matrix satisfies a given polynomial equation, this relation can be exploited to express any higher power of the matrix in terms of lower degree matrices such as A², A, and the identity matrix.

Polynomial Reduction in Matrix Computations

Using the given polynomial equation, one can perform polynomial reduction to express complex expressions like A? as a combination of A², A, and I. This technique is critical in linear algebra, especially when computing matrix powers without resorting to repeated multiplication, thereby streamlining computations and providing insights into the structure of the matrix.

Transcript

Please give Ace some feedback