If A=[ 1 2 2 2 1 2 2 2 1 ], then (a) Adj A=A-4 I3 (b) A^

step 1 So, in the given question we are told that there is a matrix that is given as A is equal to matr

If A=[ 1 2 2 2 1 2 2 2 ...

If $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$, then
(a) $\operatorname{Adj} \mathrm{A}=\mathrm{A}-4 \mathrm{I}_{3}$
(b) $\mathrm{A}^{2}=4 \mathrm{~A}-5 \mathrm{I}_{3}$
(c) $\operatorname{Adj} A=A+4 I_{3}$
(d) $A^{2}=5 A-4 I_{3}$

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Key Concepts

Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial. This theorem is a powerful tool in linear algebra as it allows one to express higher powers of a matrix in terms of lower powers and the identity matrix. It is particularly useful in deducing polynomial relationships and matrix identities that simplify the computation of matrix functions.

Matrix Polynomial Identities

Matrix polynomial identities refer to the equations that express powers of matrices, such as A², as linear combinations of lower degree matrices like A and the identity matrix. These identities, which can often be derived using the Cayley-Hamilton theorem or by direct computation, provide insight into the algebraic structure of matrices and are instrumental in simplifying complex matrix expressions and solving matrix equations.

Adjugate Matrix

The adjugate matrix of a square matrix is the transpose of its cofactor matrix. It plays a vital role in finding the inverse of a matrix since, when a matrix is invertible, its inverse can be expressed as the adjugate divided by the determinant. Additionally, it satisfies the property A × adj(A) = det(A) × I, which is useful in establishing matrix identities and relationships.

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