Question

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

Answer 1

K = -7

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