First slide

Operations of matrix

Question

Let $A = [\begin{array}{ll} 1 a \\ 0 1 \end{array}]$ , where a>0, Sum of the
series S = trace (A) + trace $(\frac{1}{2} A) +$ trace $(\frac{1}{2^{2}} A^{2}) +$ trace $(\frac{1}{2^{3}} A^{3})$ + … is

Moderate

A

3
B

4
C

6
D

8

Solution

$A = I + B$ , where $B = [\begin{array}{ll} 0 a \\ 0 0 \end{array}]$
$B^{2} = O$ we get $B^{r} = O \forall r \geq 2$
$\begin{array}{l} A^{r} = (I + B)^{r} = I + r B \\ = [\begin{array}{cc} 1 & r a \\ 0 & 1 \end{array}] \forall r \geq 1 \end{array}$
trace $(\frac{1}{2^{r}} A^{r}) = \frac{1}{2^{r}} (2) = \frac{1}{2^{r - 1}} \forall r \geq 1$ .
$S = 2 + 1 + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \dots = \frac{2}{1 - \frac{1}{2}} = 4$

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