Find A if $(3A)^T = \begin{bmatrix}1 & -1\\0 & 0\end{bmatrix}$

Question

Find $A$ when $$(3A)^T= \begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}$$

So i was given this problem and I looked through my textbook to see a practice example to follow, but I'm completely confused by it. Is there a certain formula i can apply, and how would the question differ if it was

Find $A$ when $$(3A)^T = \begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}$$

Answer 1

Hint: $(3A)^T=3A^T$, so you have $A^T$, since $(A^T)^T=A$ you have $A$.

Answer 2

$$(3A)^T= \begin{bmatrix}1 & -1\\0 & 1\end{bmatrix} \\ \iff 3A = \begin{bmatrix}1 & 0\\-1 & 1\end{bmatrix} \\ \iff A = \begin{bmatrix}\frac{1}{3} & 0\\-\frac{1}{3} & \frac{1}{3}\end{bmatrix}$$

Answer 3

Procedural Hint

Start with the outmost elements and work your way in to A.

• What property of the transpose can you use to get from $A^t$ to $A$?
• What is $cA$ equal to for some scalar c and a matrix A?