Find the transformation matrix

Algebra Level 2

Let $T: \mathbb{R}^2 \rightarrow \mathbb{R^2}$ be a linear transformation such that $T\begin{pmatrix} 2\\1 \end{pmatrix}=\begin{pmatrix} 5\\4 \end{pmatrix}$ and $T\begin{pmatrix} 3\\0 \end{pmatrix}=\begin{pmatrix} 1\\2 \end{pmatrix}$ . If a matrix $A$ is the standard matrix of $T$ , find $\det(3A)$ .

The answer is -18.

1 solution

Mark Hennings
Jun 25, 2019

We have $A\left(\begin{array}{cc} 3 & 2 \\ 0 & 1 \end{array}\right) \; = \; \left(\begin{array}{cc} 1 & 5 \\ 2 & 4 \end{array}\right)$ and hence $A \; = \; \left(\begin{array}{cc} 1 & 5 \\ 2 & 4 \end{array}\right)\left(\begin{array}{cc} 3 & 2 \\ 0 & 1 \end{array}\right)^{-1} \; = \; \tfrac13\left(\begin{array}{cc} 1 & 13 \\ 2 & 8 \end{array}\right)$ Thus $\mathrm{det}(3A) \; = \ \left| \begin{array}{cc} 1 & 13 \\ 2 & 8 \end{array}\right| \; = \; \boxed{-18}$

Find the transformation matrix

The answer is -18.

1 solution

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