Question

Find the \(2 \times 2\) matrix \(A\) such that \(A^2 = A\) and
\(A \begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.\)

Answer 1

One key thing to observe here is that \(A\) is either a singular matrix or the identity matrix :
\[A^2 = A \implies \color{red}{A^{-1}} A^2 = \color{red}{A^{-1}} A \implies A = I\]

That shows either \(A\) is the identity matrix or it is not invertible.

Answer 2

Let \(A = \begin{bmatrix} x&kx\\y&ky \end{bmatrix}\).

Use the given condition and setup two equations.