Matrices and Matrix Transformations
A matrix is a rectangular array of numbers. Matrices of the same size are added by adding corresponding entries, and multiplied by a scalar by multiplying every entry. Matrix multiplication is different: the entry in row i and column j of the product comes from multiplying row i of the first matrix by column j of the second and adding the results, which means the order matters and AB is usually not the same as BA. A 2 by 2 matrix acting on a column vector transforms the plane, and the standard transformation matrices for rotations about the origin, reflections in the axes and the lines y = x and y = -x, and enlargements centred on the origin, are worth knowing by heart.
Before you start
Make sure you're comfortable with these topics first:
Method
- Check the sizes before multiplying: the number of columns in the first matrix must equal the number of rows in the second, and the product has the rows of the first and the columns of the second.
- Multiply row by column: for each entry of the product, take the corresponding row of the first matrix and column of the second, multiply the paired numbers, and add.
- Remember that matrix multiplication is not commutative. AB and BA are generally different matrices, and one may not even be defined.
- To transform a point, write it as a column vector and multiply the matrix by that vector, with the matrix on the LEFT.
- Learn the standard 2 by 2 transformation matrices: a rotation of 90 degrees anticlockwise about the origin is [0, -1; 1, 0]; a rotation of 180 degrees is [-1, 0; 0, -1]; a reflection in the x axis is [1, 0; 0, -1]; a reflection in the y axis is [-1, 0; 0, 1]; a reflection in y = x is [0, 1; 1, 0]; and an enlargement of scale factor k centred on the origin is [k, 0; 0, k].
- To identify an unknown transformation from its matrix, apply it to the points (1, 0) and (0, 1) and see where those two unit vectors land, since their images are exactly the columns of the matrix.
Worked example
A = [2, 1; 0, 3] and B = [1, 4; 2, -1]. Find AB, and describe the transformation represented by the matrix [0, -1; 1, 0].
- Top-left entry of AB: row 1 of A times column 1 of B = (2)(1) + (1)(2) = 4.
- Top-right entry: row 1 of A times column 2 of B = (2)(4) + (1)(-1) = 8 - 1 = 7.
- Bottom-left entry: row 2 of A times column 1 of B = (0)(1) + (3)(2) = 6.
- Bottom-right entry: row 2 of A times column 2 of B = (0)(4) + (3)(-1) = -3. So AB = [4, 7; 6, -3].
- For the transformation, apply the matrix to (1, 0): it gives the first column, (0, 1). Apply it to (0, 1): it gives the second column, (-1, 0).
- The point one unit along the x axis has moved to one unit up the y axis, and the point one unit up the y axis has moved to one unit along the negative x axis. That is a rotation of 90 degrees anticlockwise about the origin.
Practice questions
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Q1Find [1, 2; 3, 4] + [5, 0; -1, 2].Show answer
Answer: [6, 2; 2, 6].
Q2Find 3 times the matrix [2, -1; 0, 4].Show answer
Answer: [6, -3; 0, 12].
Q3Why is AB usually different from BA for matrices?Show answer
Answer: Because each entry of a product comes from a row of the first matrix and a column of the second, so swapping the order pairs different numbers together.
Q4Which matrix represents a reflection in the x axis?Show answer
Answer: [1, 0; 0, -1].
Q5Apply the matrix [2, 0; 0, 2] to the point (3, -1).Show answer
Answer: (6, -2), an enlargement of scale factor 2 centred on the origin.
Q6A 2 by 3 matrix is multiplied by a 3 by 4 matrix. What size is the product?Show answer
Answer: 2 by 4.
Q7How can you identify the transformation a 2 by 2 matrix represents?Show answer
Answer: Its columns are the images of (1, 0) and (0, 1), so reading the columns tells you where those unit vectors go.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
P = [3, 2; 1, 0] and Q = [-1, 4; 2, 5]. (a) Work out PQ. (b) Work out QP. (c) State what your answers show about matrix multiplication.
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The matrix M = [0, 1; 1, 0] is applied to the triangle with vertices A(1, 0), B(3, 0) and C(3, 2). Find the images of the three vertices and describe the single transformation M represents.
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Free printable worksheet
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