Core Pure: Matrices and Transformations Depth
In Core Pure, part of A-level Further Maths, matrices and transformations depth extends 2D transformation matrices into three dimensions, and uses eigenvalues together with eigenvectors rather than eigenvalues alone. Standard 3 by 3 matrices represent a rotation about a coordinate axis or a reflection in a coordinate plane, and combine by matrix multiplication in the order the transformations are applied. Separately, once a matrix M has two distinct eigenvalues, writing M = PDP^-1 (with P holding the eigenvectors as columns and D holding the matching eigenvalues on the diagonal, rows separated by a semicolon in this notation) makes a high power M^n quick to find, since D^n just raises each diagonal entry to the power n.
Method
- Learn the standard 3 by 3 matrix for a rotation about a coordinate axis, for example rotation about the z-axis through angle theta is (cos theta -sin theta 0; sin theta cos theta 0; 0 0 1), and for a reflection in a coordinate plane, for example reflection in the plane y = 0 is (1 0 0; 0 -1 0; 0 0 1).
- Find the determinant of a 3 by 3 matrix by expanding along a row or column using 2 by 2 minors with alternating signs, and interpret det(M) as the volume scale factor of the transformation, negative if it reverses orientation.
- To find an eigenvalue, solve the characteristic equation det(M - lambda I) = 0; to find its eigenvector, substitute that eigenvalue back into (M - lambda I)v = 0 and solve for a nonzero vector v (any nonzero scalar multiple of v is also valid).
- Once two distinct eigenvalues and their eigenvectors are known for a 2 by 2 matrix, form P with the eigenvectors as its columns and D as the diagonal matrix of the matching eigenvalues, in the same order, so that M = PDP^-1.
- Use M^n = PD^nP^-1 to compute a high power of M efficiently, since D^n is simply each diagonal entry of D raised to the power n.
- For a composite 3D transformation, multiply the individual matrices in the order the transformations are applied, with the matrix for the transformation applied first placed on the right, next to the column vector.
- Check a transformation matrix by applying it to a simple test point and confirming the image matches the expected geometric effect, or check a diagonalisation by multiplying PDP^-1 back out and confirming it returns the original matrix M.
Worked example
The matrix M = (5 4; 1 2) represents a transformation. (a) Find the eigenvalues and corresponding eigenvectors of M. (b) Hence write M in the form M = PDP^-1, and use this to find M^4.
- Find the eigenvalues by solving det(M - lambda I) = 0: (5-lambda)(2-lambda) - 4(1) = 0 expands to lambda^2 - 7lambda + 6 = 0, which factorises to (lambda-1)(lambda-6) = 0, giving lambda = 1 or lambda = 6.
- For lambda = 1, solve (M - I)v = 0: the matrix (4 4; 1 1) gives x + y = 0, so an eigenvector is (1,-1). For lambda = 6, solve (M - 6I)v = 0: the matrix (-1 4; 1 -4) gives x = 4y, so an eigenvector is (4,1).
- Form P = (1 4; -1 1) (eigenvectors as columns) and D = (1 0; 0 6) (matching eigenvalues on the diagonal), so that M = PDP^-1.
- Find P^-1: det(P) = (1)(1) - (4)(-1) = 5, so P^-1 = (1/5) times (1 -4; 1 1).
- Use M^4 = PD^4P^-1 with D^4 = (1 0; 0 1296) since 6^4 = 1296: multiplying P by D^4 gives (1 5184; -1 1296), then multiplying this by P^-1 gives (1/5) times (5185 5180; 1295 1300).
- Final answer: eigenvalues 1 and 6 with eigenvectors (1,-1) and (4,1) respectively; M^4 = (1037 1036; 259 260).
Practice questions
Try each question, then tap to reveal the answer.
Q1Write down the 3 by 3 matrix representing a rotation about the z-axis through angle theta.Show answer
Answer: (cos theta -sin theta 0; sin theta cos theta 0; 0 0 1)
Q2Write down the 3 by 3 matrix representing a reflection in the plane x = 0 (the yz-plane).Show answer
Answer: (-1 0 0; 0 1 0; 0 0 1), since reflection in x = 0 sends (x,y,z) to (-x,y,z).
Q3Find det(N) where N = (2 0 1; 1 3 0; 0 2 4), by expanding along the first row.Show answer
Answer: 26, from 2 times det(3 0; 2 4) - 0 + 1 times det(1 3; 0 2) = 2(12-0) + 1(2-0) = 24+2 = 26.
Q4The matrix M has eigenvalues 2 and 5. State the two eigenvalues of M^3.Show answer
Answer: 8 and 125 (2^3 and 5^3), since if Mv = lambda*v then M^3v = lambda^3*v for the same eigenvector v.
Q5Find the eigenvalues of the matrix (3 2; 2 3) by solving its characteristic equation.Show answer
Answer: lambda = 1 or lambda = 5, from (3-lambda)^2 - 4 = 0, giving lambda^2 - 6lambda + 5 = 0 which factorises to (lambda-1)(lambda-5) = 0.
Q6Find an eigenvector corresponding to the eigenvalue lambda = 5 for the matrix (3 2; 2 3).Show answer
Answer: (1,1), from solving (-2 2; 2 -2)v = 0, which gives -2x + 2y = 0, so x = y.
Q7A transformation consists of a rotation about the z-axis through 90 degrees, followed by a reflection in the xy-plane. State the order in which the two 3 by 3 matrices should be multiplied to represent the combined transformation, applied to a column vector v.Show answer
Answer: The reflection matrix multiplied by the rotation matrix, i.e. (Reflection)(Rotation)v, since the matrix for the transformation applied first (the rotation) goes on the right, next to v.
Q8A 2 by 2 matrix M can be written as M = PDP^-1 with D = (4 0; 0 -1). Explain why the eigenvalues of M^2 are both positive.Show answer
Answer: M^2 = PD^2P^-1 and D^2 = (16 0; 0 1), so the eigenvalues of M^2 are 4^2 = 16 and (-1)^2 = 1, both positive because squaring removes any negative sign.
Exam-style questions
Written in the style of a A Level Further Maths exam paper, with a full mark scheme.
The matrix A = (6 -4; 3 -1) represents a transformation. (a) Show that the eigenvalues of A are 2 and 3. (b) Find an eigenvector corresponding to each eigenvalue.
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A transformation T of 3D space consists of a reflection in the plane z = 0, followed by a rotation about the z-axis through 90 degrees anticlockwise (viewed from the positive z-axis looking towards the origin). (a) Write down the 3 by 3 matrix for each transformation. (b) Find the matrix representing the combined transformation T. (c) Hence find the image of the point (2,1,3) under T.
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The matrix B = (2 1 0; 0 3 1; 1 0 2) represents a transformation of 3D space. (a) Find det(B). (b) A solid has volume 15 cm^3. Find the volume of its image under the transformation represented by B.
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