Question

If A = [
1 3
3 4
] and A
2 − kA – 5I = 0 then k = _____

Answer 1

Given: A = $\left[\begin{array}{ccc}1&3\\3&4\end{array}\right]$ and A²-kA -5I = 0

To find: Value of k?

Solution:

• Now we have given the matrix A =$\left[\begin{array}{ccc}1&3\\3&4\end{array}\right]$, so A² will be:

                   A² = $\left[\begin{array}{ccc}1&3\\3&4\end{array}\right]$x $\left[\begin{array}{ccc}1&3\\3&4\end{array}\right]$

                        = $\left[\begin{array}{ccc}10&15\\15&25\end{array}\right]$

• Now putting values of A and A² in the given equation, we get:

                  A²-kA -5I = 0

                  $\left[\begin{array}{ccc}10&15\\15&25\end{array}\right]$  - $\left[\begin{array}{ccc}k&3k\\3k&4k\end{array}\right]$  -  $\left[\begin{array}{ccc}5&0\\0&5\end{array}\right]$ = $\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$

• So the following equation is found:

                  10 - k - 5 = 0

• Solving this, we get:

                  5 = k

Answer:

               So the value of k is 5.

Answer 2

Given:

  $A = \left[\begin{array}{cc}1&3\\3&4\end{array}\right]$

       

$A^{2} - kA - 5I = 0$

To Find:

Find the value of k.

Solution:

A  is a 2×2 matrix

$a_{11}= 1 \\a_{12} = 3 \\a_{21}= 3 \\a_{22} = 4$

We have to satisfy the given polynomial to find the value of k.

Matrix multiplication :

$A^{2} =$ A×A

$A^{2} = \left[\begin{array}{cc}1&3\\3&4\end{array}\right]\left[\begin{array}{ccc}1&3\\3&4\\\end{array}\right]$

$A^{2} = \left[\begin{array}{cc}10&15\\15&25\\\end{array}\right]$

Substitute in the given equation;

$\left[\begin{array}{cc}10&15\\15&25\\\end{array}\right] -\left[\begin{array}{cc}k&3k\\3k&4k\\\end{array}\right] - \left[\begin{array}{ccc}5&0\\0&5\\\end{array}\right] = 0$

$\left[\begin{array}{cc}10&15\\15&25\\\end{array}\right]= \left[\begin{array}{cc}k&3k\\3k&4k\\\end{array}\right] + \left[\begin{array}{ccc}5&0\\0&5\\\end{array}\right]$

Solve the linear equations to calculate the value of k;

$10 = k + 5\\ k = 5$

The value of K to satisfy the above equation is k = 5