If`D=d i ag[d_1, d_2, d_n]` , then prove that `f(D)=d i ag[f(d_1),f(d_2), ,f(d_n)],w h e r ef(x)` is a polynomial with scalar coefficient.

If D=diag(d_1,d_2,d_3,…,d_n)" where "d ne 0" for all " I = 1,2,…,n," then " D^(-1) is equal to

If D=diag(d_1,d_2,d_3,…,d_n)" where "d ne 0" for all " I = 1,2,…,n," then " D^(-1) is equal to

Let D_1=|a b a+b c d c+d a b a-b|a n dD_2=|a c a+c b d b+d a c a+b+c| then the value of |(D_1)/(D_2)|,w h e r eb!=0a n da d!=b c , is _____.

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D^*F(x) by the formula D*f(x)=lim_(h->0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x) mean [f(x)]^2 and if f(x)=xlogx ,then D^*f(x)|_(x=e) has the value (A)e (B) 2e (c) 4e (d) none of these

In Figure, D E || A C and D F || A E . Prove that (E F)/(B F)=(E C)/(B E) .

In Fig. if E F|| D C|| A Bdot Prove that (A E)/(E D)=(B F)/(F C) . (FIGURE)

In Fig. if E F|| D C|| A Bdot Prove that (A E)/(E D)=(B F)/(F C) . (FIGURE)

Let f(x)=x+f(x-1) for AAx in R . If f(0)=1,f i n d \ f(100) .

If A=d i ag\ (a\ \ b\ \ c) , show that A^n=d i ag\ (a^n\ \ b^n\ \ c^n) for all positive integer n .

If A=d i ag(abc), show that A^n=d i ag(a^nb^nc^n) for all positive integer n .