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 = [tex]\left[\begin{array}{ccc}1&0\\-1&7\\\end{array}\right][/tex]
We are to find K if =0
A² = [tex]\left[\begin{array}{ccc}1&0\\-1&7\\\end{array}\right][/tex][tex]\left[\begin{array}{ccc}1&0\\-1&7\\\end{array}\right][/tex]
A² = [tex]\left[\begin{array}{ccc}1&0\\-8&49\\\end{array}\right][/tex]
8A = 8[tex]\left[\begin{array}{ccc}1&0\\-1&7\\\end{array}\right][/tex]
8A = [tex]\left[\begin{array}{ccc}8&0\\-8&56\\\end{array}\right][/tex]
Since [tex]I = \left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right] \\[/tex]
[tex]KI = \left[\begin{array}{ccc}K&0\\0&K\\\end{array}\right][/tex]
Substitute the resulting matrices into the expression above:
A^2-8A-KI = [tex]\left[\begin{array}{ccc}1&0\\-8&49\\\end{array}\right][/tex] - [tex]\left[\begin{array}{ccc}8&0\\-8&56\\\end{array}\right][/tex] - [tex]\left[\begin{array}{ccc}K&0\\0&K\\\end{array}\right][/tex] =[tex]\left[\begin{array}{ccc}0&0\\0&0\\\end{array}\right][/tex]
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
