If […] =Aλ^2+Bλ+C, where A, B and C are matrices then find matrix B.

We have $\begin{bmatrix}λ^2-2λ+1&λ-2\\1-λ^2+3λ&1-λ^2\end{bmatrix}=Aλ^2+Bλ+C$

Putting $λ = 0$, we get

$C=\begin{bmatrix}1&-2\\1&1\end{bmatrix}$

Putting $λ =1$, we get

$A+B+C=\begin{bmatrix}0&-1\\3&0\end{bmatrix}$   ...(1)

Putting $λ =-1$, we get

$A-B+C=\begin{bmatrix}4&-3\\-3&0\end{bmatrix}$   ...(2)

Subtracting (2) from (1), we get

$2B=\begin{bmatrix}0&-1\\3&0\end{bmatrix}-\begin{bmatrix}4&-3\\-3&0\end{bmatrix}=\begin{bmatrix}-4&2\\6&0\end{bmatrix}$

$∴B=\begin{bmatrix}-2&1\\3&0\end{bmatrix}$