Question

If A=[4325], find x and y such that A2−xA+yI=0.

Answer 1

Given: The matrix $\left[\begin{array}{cc}4&3\\2&5&\end{array}\right]$ and the equation A^2 − xA + yI = 0.

To find: The value of x and y?

Solution:

• Now we have given the matrix $\left[\begin{array}{cc}4&3\\2&5\end{array}\right]$.

• Now A^2 will be: $\left[\begin{array}{ccc}4&3\\2&5\end{array}\right] \left[\begin{array}{ccc}4&3\\2&5\end{array}\right] = \left[\begin{array}{ccc}22&27\\18&31\end{array}\right]$

• Putting the values in the equation A^2 − xA + yI = 0, we get:

           $\left[\begin{array}{ccc}22&27\\18&31\end{array}\right] - \left[\begin{array}{ccc}4x&3x\\2x&5x\end{array}\right] + \left[\begin{array}{ccc}y&0\\0&y\end{array}\right] = \left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$

• Now the equations are:

          22 - 4x + y = 0 .............(i)

          27 - 3x = 0 ................(ii)

          18 - 2x = 0  ..............(iii)

          31 - 5x + y = 0 ........ .....(iv)

• From (ii), we get:

          27 = 3x

          x = 9

• Putting x = 9 in (iv), we get:

         31 - 45 + y = 0

         -14 + y = 0

         y = 14

Answer:

           So the value of x is 9 and y is 14.