Paper 1, Section II, A

Vectors and Matrices | Part IA, 2021

(a) For an n×nn \times n matrix AA define the characteristic polynomial χA\chi_{A} and the characteristic equation.

The Cayley-Hamilton theorem states that every n×nn \times n matrix satisfies its own characteristic equation. Verify this in the case n=2n=2.

(b) Define the adjugate matrix adj(A)\operatorname{adj}(A) of an n×nn \times n matrix AA in terms of the minors of AA. You may assume that

Aadj(A)=adj(A)A=det(A)IA \operatorname{adj}(A)=\operatorname{adj}(A) A=\operatorname{det}(A) I

where II is the n×nn \times n identity matrix. Show that if AA and BB are non-singular n×nn \times n matrices then

adj(AB)=adj(B)adj(A)\operatorname{adj}(A B)=\operatorname{adj}(B) \operatorname{adj}(A)

(c) Let MM be an arbitrary n×nn \times n matrix. Explain why

(i) there is an α>0\alpha>0 such that MtIM-t I is non-singular for 0<t<α0<t<\alpha;

(ii) the entries of adj(MtI)\operatorname{adj}(M-t I) are polynomials in tt.

Using parts (i) and (ii), or otherwise, show that ()(*) holds for all matrices A,BA, B.

(d) The characteristic polynomial of the arbitrary n×nn \times n matrix AA is

χA(z)=(1)nzn+cn1zn1++c1z+c0\chi_{A}(z)=(-1)^{n} z^{n}+c_{n-1} z^{n-1}+\cdots+c_{1} z+c_{0}

By considering adj (AtI)(A-t I), or otherwise, show that

adj(A)=(1)n1An1cn1An2c2Ac1I.\operatorname{adj}(A)=(-1)^{n-1} A^{n-1}-c_{n-1} A^{n-2}-\cdots-c_{2} A-c_{1} I .

[You may assume the Cayley-Hamilton theorem.]

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