Verify that the matrix \(M\) in the proof of Theorem 2.6 and Corollary 2.7 is a lower triangular matrix with entries 1 on and below the diagonal. Show that the inverse matrix \(M^{-1}\) is a lower triangular matrix with entries 1 on the diagonal and -1 directly below the diagonal.

Data From Theorem 2.6

Data From Corollary 2.7

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2.6 Theorem. A one-dimensional Brownian motion (B)to is a Gaussian process. For to 0 1, the vector I = (Bt,..., Bt,) is a Gaussian random variable with mean vector m = 0 = R" and a strictly positive definite, symmetric Ex. 2.5 covariance matrix C = (tj ^ tk)j,k=1...,n: Ee(D) = (c) e Moreover, the probability distribution of I is given by (2.9) P(I dx)= 1 (2)/2 det C 1 exp ((x, cx)) dx (2.10a) 1 1(x-x1-1) exp dx (2.10b) tj- tj-1 (2)/2 (t-t-1) with x = (x1,...,xn) e R" and xo := 0. Proof. Set A = (Bt, - Bto, Btz - Bt,..., Bt. - Bt) and observe that we can write B(tk)-B(to)=(Bt, - Bt,-1). Thus, r = ) = MA