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 Elements: Identify the matrix and its elements. $\newline$ The given matrix is: $\newline$ $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ $\newline$ This is a $2$ x $2$ matrix with elements a $11$ = $-1$ , a $12$ = $0$ , a $21$ = $0$ , and a $22$ = $1$ .
Effect on Basis Vectors: Analyze the effect of the matrix on the basis vectors. $\newline$ The effect of the matrix on the x-axis basis vector ( $1$ , $0$ ) is: $\newline$ $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \end{bmatrix}$ $\newline$ This means that the x-axis basis vector is reflected across the y-axis.
Effect on x-axis: Analyze the effect of the matrix on the y-axis basis vector. $\newline$ The effect of the matrix on the y-axis basis vector ( $0$ , $1$ ) is: $\newline$ $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ $\newline$ This means that the y-axis basis vector remains unchanged.
Effect on y-axis: Determine the geometric effect of the transformation. $\newline$ Since the $x$ -axis basis vector is reflected across the $y$ -axis and the $y$ -axis basis vector remains unchanged, the overall effect of the transformation is a reflection across the $y$ -axis.