Practicing Success
If $\begin{bmatrix}λ^2-2λ+1&λ-2\\1-λ^2+3λ&1-λ^2\end{bmatrix}=Aλ^2+Bλ+C$, where A, B and C are matrices then find matrix B. |
$\begin{bmatrix}-2&1\\-3&0\end{bmatrix}$ $\begin{bmatrix}-2&-1\\3&0\end{bmatrix}$ $\begin{bmatrix}-2&1\\3&0\end{bmatrix}$ $\begin{bmatrix}-2&1\\-3&1\end{bmatrix}$ |
$\begin{bmatrix}-2&1\\3&0\end{bmatrix}$ |
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 $2B=\begin{bmatrix}0&-1\\3&0\end{bmatrix}-\begin{bmatrix}4&-3\\-3&0\end{bmatrix}=\begin{bmatrix}-4&2\\6&0\end{bmatrix}$ $∴B=\begin{bmatrix}-2&1\\3&0\end{bmatrix}$ |