Question

Let A be a 2 x 2 matrix. Show that some non-trivial linear combination of A^4, A^3, A^2, A. and I2 is equal 0. Generalize to n x n matrices. Note that I2 is 2 x 2 identity matrix.

Answer 1

Let $A=I=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ $(2\times 2 \ matrix)$

$\therefore A^4=A^3=A^2=I=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Linear Combination:

$aA^4+bA^3+cA^2+dA+eI^2=0$

$\Rightarrow (a+b+c+d+e)\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=0$

$\therefore a+b+c+d+e=0$

Now, solving for $(n\times n \ matrix):$

Let $A= I_n=\begin{bmatrix} 1 & 0 & 0& 0& 0& 0& 0 .......& 0\\ 0 & 1 & 0& 0& 0& 0& 0.......& 0\\ 0 & 0 & 1& 0& 0& 0& 0.......& 0\\ . \\ . \\ . \\ . 0 & 0 & 0& 0& 0& 0& 0.......& 1\\ \end{bmatrix}$

$\therefore A^4=A^3=A^2=A= I_n=\begin{bmatrix} 1 & 0 & 0& 0& 0& 0& 0 .......& 0\\ 0 & 1 & 0& 0& 0& 0& 0.......& 0\\ 0 & 0 & 1& 0& 0& 0& 0.......& 0\\ . \\ . \\ . \\ . 0 & 0 & 0& 0& 0& 0& 0.......& 1\\ \end{bmatrix}$

Linear Combination:

$aA^4+bA^3+cA^2+dA+eI_n^2=0$

$\Rightarrow (a+b+c+d+e)\begin{bmatrix} 1 & 0 & 0& 0& 0& 0& 0 .......& 0\\ 0 & 1 & 0& 0& 0& 0& 0.......& 0\\ 0 & 0 & 1& 0& 0& 0& 0.......& 0\\ . \\ . \\ . \\ . 0 & 0 & 0& 0& 0& 0& 0.......& 1\\ \end{bmatrix}=0$

$\therefore a+b+c+d+e=0$