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Matrix algebra and transformations questions
Grids of numbers with their own arithmetic, and a second life as movements of space. Rotations, reflections, stretches and enlargements all fit in a handful of entries.
7 original questions · 27 marks · the matrix algebra and transformations notes · Matrices
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Write down the 2 × 2 matrices for a rotation of 90° anticlockwise about the origin and for a reflection in the line y = x.
Worked answer
Rotation: rows (0, −1) and (1, 0). Reflection in y = x: rows (0, 1) and (1, 0). B1 B1 for the two matrices. Both can be read off from where (1, 0) and (0, 1) land, since the images are the columns.Let A have rows (2, 1), (0, 3) and B have rows (1, −1), (2, 0). Find AB and BA, and comment.
Worked answer
AB has rows (4, −2), (6, 0); BA has rows (2, −2), (4, 2). They differ, because matrix multiplication is not commutative, so the order of a product must never be swapped mid-working. M1 for a correct multiplication method, A1 for both products, B1 for the comment on commutativity.Find the image of the point (3, 1) under the rotation of 90° anticlockwise about the origin.
Worked answer
Multiply: rows (0, −1), (1, 0) applied to (3, 1) give (−1, 3). B1 for the rotation matrix, M1 for the multiplication, A1 for (−1, 3). A sketch confirms it. The point swings from the first quadrant into the second, a quarter turn against the clock.A transformation reflects in the x-axis and then rotates 90° anticlockwise. Find the single matrix representing the combined transformation and identify it.
Worked answer
The reflection acts first, so it sits on the right of the product. RM is rows (0, −1), (1, 0) times rows (1, 0), (0, −1), which gives rows (0, 1), (1, 0). That is the reflection in y = x. B1 for the reflection matrix, M1 for multiplying in the order RM, A1 for rows (0, 1), (1, 0), A1 for naming the reflection in y = x. Writing MR instead gives rows (0, −1), (−1, 0), the reflection in y = −x, so the order genuinely matters.A linear transformation sends (1, 0) to (2, 0) and (0, 1) to (1, 3). Write down its matrix, and find the image of (2, 5).
Worked answer
The images of the base vectors are the columns, giving rows (2, 1), (0, 3). Then (2, 5) maps to (2 × 2 + 1 × 5, 0 + 3 × 5) = (9, 15). B1 for the matrix, M1 for the multiplication, A1 for (9, 15). Columns-are-images turns matrix building into reading the question.Using the matrix with rows (0, −1), (1, 0), show that applying the 90° rotation four times returns every point to where it started, and explain what this says about the matrix.
Worked answer
Squaring gives rows (−1, 0), (0, −1), which is −I, a half turn. Squaring again gives (−I)² = I, so R⁴ = I. M1 for squaring, A1 for −I, M1 for squaring again, A1 for the fourth power being the identity. Four quarter turns make a full turn, and the matrix records that as its fourth power being the identity.R is a rotation of 90° anticlockwise about the x-axis and S is a rotation of 90° anticlockwise about the z-axis, both viewed from the positive end of the axis. Find the matrix representing S followed by R, use it to find the image of (1, 2, 3), and show that R followed by S is a different transformation.
Worked answer
Build each matrix from the images of the base vectors. R fixes i, sends j to k and k to −j, so R has rows (1, 0, 0), (0, 0, −1), (0, 1, 0). S fixes k, sends i to j and j to −i, so S has rows (0, −1, 0), (1, 0, 0), (0, 0, 1). S first means S on the right, so the combined matrix is RS, with rows (0, −1, 0), (0, 0, −1), (1, 0, 0). Applying it to (1, 2, 3) gives (−2, −3, 1). M1 for building a matrix from the images of the base vectors, A1 for R, A1 for S, M1 for the product in the order RS, A1 for RS, A1 for (−2, −3, 1), M1 for reversing the order, A1 for SR differing. Reversing the order gives SR, with rows (0, 0, 1), (1, 0, 0), (0, 1, 0), which sends (1, 2, 3) to (3, 1, 2). The two matrices differ, so the transformations differ. Rotations about different axes generally do not commute, and these two 90° rotations give different products, which is easy to confirm with a die or a book held at arm's length.
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