(a) Prove that for any 2 Atilde 2 matrix T there are scalars c0, , c4 that are not all 0 such that the combination c4T4 c3T3 c2T2 c1T c0I is the zero matrix (where I is the 2 Atilde 2 identity matrix, with 1 's in its 1, 1 and 2, 2 entries and zeroes elsewhere, see Exercise 34) (b) Let p(x) be a polynomial p(x) cnxn c1x c0 If T is a square matrix we define p(T) to be the matrix cnTn c1T c0I (where I is the appropriately sized identity matrix) Prove that for any square matrix there is a polynomial such that p(T) is the zero matrix (c) The minimal polynomial m(x) of a square matrix is the polynomial of least degree, and with leading coefficient 1, such that m(T) is the zero matrix Find the minimal polynomial of this matrix (This is the representation with respect to Icirc micro 2, Icirc micro 2, the standard basis, of a rotation through Iuml 6 radians counterclockwise ) V3 2 1 2 1 2 B 2,

Question

(a) Prove that for any 2  Atilde  2 matrix T there are scalars c0,       , c4 that are not all 0 such that the combination c4T4   c3T3   c2T2   c1T   c0I is the zero matrix (where I is the 2  Atilde  2 identity matrix, with 1 's in its 1, 1 and 2, 2 entries and zeroes elsewhere, see Exercise 34)  (b) Let p(x) be a polynomial p(x)   cnxn           c1x   c0  If T is a square matrix we define p(T) to be the matrix cnTn           c1T   c0I (where I is the appropriately sized identity matrix)  Prove that for any square matrix there is a polynomial such that p(T) is the zero matrix  (c) The minimal polynomial m(x) of a square matrix is the polynomial of least degree, and with leading coefficient 1, such that m(T) is the zero matrix  Find the minimal polynomial of this matrix  (This is the representation with respect to  Icirc  micro 2,  Icirc  micro 2, the standard basis, of a rotation through  Iuml  6 radians counterclockwise ) V3 2 1 2 1 2 B 2,

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a The vector space M 22 has dimension four The set T 4 T I has five elements and thus is li...

Question: (a) Prove that for any 2 Ã 2 matrix T there are scalars c0, . . . , c4 that are not all 0 such

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