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nadya68 [22]
3 years ago
13

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

Mathematics
1 answer:
Snezhnost [94]3 years ago
8 0

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

<em>k = -7</em>

<em>Hence the value of K is -7</em>

<em></em>

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