Consider this matrix transformation: $\newline$ $\left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right]$ $\newline$ What is the geometric effect of this transformation? $\newline$ Choose $1$ answer: $\newline$ (A) A reflection across the $y$ axis $\newline$ (B) A reflection across the line $y=x$ $\newline$ (C) A rotation about the origin by $90^{\circ}$ $\newline$ (D) A rotation about the origin by $180^{\circ}$

Full solution

Q. Consider this matrix transformation:

\newline

\left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right]

\newline

What is the geometric effect of this transformation?

\newline

Choose

1

answer:

\newline

(A) A reflection across the

y

axis

\newline

(B) A reflection across the line

y=x

\newline

90^{\circ}

\newline

(D) A rotation about the origin by

180^{\circ}

Identify matrix transformation: Identify the matrix transformation. $\newline$ The given matrix is: $\newline$ \left[\begin{array}{cc}$\newline-1 & 0 (\newline$0 & -1 $\newline$ \end{array}\right]\)
Effect on basis vectors: Understand the effect of the matrix on the basis vectors. The effect of the matrix on the basis vectors $i (1,0)$ and $j (0,1)$ will tell us the geometric transformation. Multiplying the matrix by $i$ : $\begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 1 \ 0 \end{bmatrix} = \begin{bmatrix} -1 \ 0 \end{bmatrix}$
Effect on basis vector $j$ : Understand the effect of the matrix on the basis vector $j$ . $\newline$ Multiplying the matrix by $j$ : $\newline$ $\begin{bmatrix}-1 & 0\ 0 & -1\end{bmatrix} \times \begin{bmatrix}0\ 1\end{bmatrix} = \begin{bmatrix}0\ -1\end{bmatrix}$
Analysis of transformation results: Analyze the results of the transformation on the basis vectors. $\newline$ The basis vector $i$ $(1,0)$ is transformed to $(-1,0)$ , and the basis vector $j$ $(0,1)$ is transformed to $(0,-1)$ . This means that the $x$ -coordinate of any point is negated, and the $y$ -coordinate of any point is also negated.
Geometric effect of negating coordinates: Determine the geometric effect of negating both coordinates. Negating both the $x$ and $y$ coordinates of every point in the plane results in a reflection across the origin, which is equivalent to a rotation about the origin by $180$ degrees.