Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.

Prove that the null space of an mxn matrix is a sub space of R^n

10. Show that if 0 (18), M₂ - (89) = then the span of {M₁, M2, M3} is the set of all symmetric 2×2 matrices. M₁ = 2 and_M3 = ( - (3) 2

Suppose A = {a1, . . . , am} and B = {b1, . . . , bn}. How many relations are therefrom A to B? Hint: Count the number of m × n matrices with (0,1)-entries.If R is a relation from A to B we define the inverse relation R^-1 from B to A by bR^-1a whenever aRb.

The symmetric matrix corresponding to the quadratic form X1 x2 = x} – 2x1x2 + 2x1x3 – 3x + 4x2x3 – 5x X3 takes the form a b. d e C e f, for some real numbers а, b, с, d, e,f. Enter these values, in order. A

(a) Show that the following set of matrices is closed under linear combina- tions. a=b): a,bER}. i.e. Show that c₁M₁ + c₂M₂ € S₁ for all choices M₁, M₂ € S₁ and all scalars C₁, C₂ € R. S₁ is {(ª + ² a + 2b a+b a - 3b (b) Show that the following set of matrices is closed under affine combina- tions. S3 = a -{(82): QbER}. a, S₂ == i.e. Show that c₁ M₁+c₂M₂ € S₂ for all choices M₁, M₂ € S₂ and C₁, C₂ € R satisfying c₁ + C₂ = 1. (c) Show that the following set of matrices is not closed under affine combi- nations. {(66%) -2²=0}. : x,y ER, y - x²

Help with number 4 please, a and b

Suppose that A is an nxn invertible matrix. Prove the following statements:  (a) If there exists an nxn matrix D such that AD=I_n then D=A^-1. (b) If B and C are matrices such that A(B-C)=0 (where 0 is a zero matrix of appropriate size), then B=C.  Hint: Both of these are simplier to prove using matrix algebra rather than the definition of matrix multiplication.  I_n is the identity matrix and A^-1 is A inverse matrix.

2. Let G be the following graph. V1 V2 V3 V5 VA (a) Let A be the adjacency matrix of G. Find A. (b) Using only A and its powers, determine how many walks of length EXACTLY 3 are there starting at 2 and ending at v4. Explain. (c) Using only A and its powers, determine how many walks of length AT MOST 3 are there starting at 2 and ending at v4. Explain.

Let A be the adjacency matrix of a complete graph K4.(a) Write down matrix A.(b) How many possible walks with length 2 are there from a (any) node to itself (e.g., from Node 2 to itself) (c) How many possible walks with length 3 are there from a (any) node to the other node (e.g., from Node 2 to Node 3 or Node 1 to Node 4)