Question

If A=[1 0 -1 7] find K such that A^2-8A-ki-KI=0

Answer 1

Answer:

K = -7

Step-by-step explanation:

If A=[1 0 -1 7] find K such that A^2-8A-ki-KI=0

Given the 2×2 matrices

A = $\left[\begin{array}{ccc}1&0\\-1&7\\\end{array}\right]$

We are to find K if  =0

A² =  $\left[\begin{array}{ccc}1&0\\-1&7\\\end{array}\right]$$\left[\begin{array}{ccc}1&0\\-1&7\\\end{array}\right]$

A² = $\left[\begin{array}{ccc}1&0\\-8&49\\\end{array}\right]$

8A = 8$\left[\begin{array}{ccc}1&0\\-1&7\\\end{array}\right]$

8A = $\left[\begin{array}{ccc}8&0\\-8&56\\\end{array}\right]$

Since $I = \left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right]$

$KI = \left[\begin{array}{ccc}K&0\\0&K\\\end{array}\right]$

Substitute the resulting matrices into the expression above:

A^2-8A-KI =  $\left[\begin{array}{ccc}1&0\\-8&49\\\end{array}\right]$ - $\left[\begin{array}{ccc}8&0\\-8&56\\\end{array}\right]$ - $\left[\begin{array}{ccc}K&0\\0&K\\\end{array}\right]$ =$\left[\begin{array}{ccc}0&0\\0&0\\\end{array}\right]$

From the expression, we have the equations;

1 - 8 - k = 0

-7 - k = 0

-7 = k

k = -7

Hence the value of K is -7