Question

for the above matrix by using cayley hamilton theorem find A+
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Answer 1

Answer:

A matrix is a rectangular array of numbers (or other mathematical objects) for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars, each of which is a member of F.

Step-by-step explanation:

Answer 2

Answer:

The Cayley–Hamilton theorem states that substituting the matrix A for x in polynomial, p(x) = det(xIn – A), results in the zero matrices, such as:

p(A) = 0

It states that a ‘n x n’ matrix A is demolished by its characteristic polynomial det(tI – A), which is monic polynomial of degree n. The powers of A, found by substitution from powers of x, are defined by recurrent matrix multiplication; the constant term of p(x) provides a multiple of the power A0, where power is described as the identity matrix.

The theorem allows An to be articulated as a linear combination of the lower matrix powers of A. If the ring is a field, the Cayley–Hamilton theorem is equal to the declaration that the smallest polynomial of a square matrix divided by its characteristic polynomial.

Example of Cayley-Hamilton Theorem

1.) 1 x 1 Matrices

For 1 x 1 matrix A(a1,1) the characteristic polynomial is given by p(λ)=λ–a and so

p(A) = (a) – (a1,1) = 0 is obvious.

2.) 2 x 2 Matrices

Let us look this through an example

A = (1324)

p(λ)=det(λI2−A)=det(λ−1−3−2λ−4)=(λ−1)(λ−4)−(−2)(−3)=λ2−5λ−2

The Cayley-Hamilton claims that if, we define

p(X) = X2−5X−2I2

then,

p(A) = A2−5A−2I2 = (0000)

We can verify this result by computation

A2−5A−2I2 = (7151022)−(5151020)−(2002)=(0000)

For a generic 2 x 2 matrix,

A=(acbd)

the resultant polynomial is given by P(λ)=λ2−(a+d)λ+(ad−bc) , so the Cayley-Hamilton theorem states that

p(A)=A2−(a+d)A+(ad−bc)I2=(0000)

it is always the case, which is evident by working out on A2