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Matrix Multiplication

How to multiply a matrix by a scalar, a column vector or another matrix, and use the identity matrix and matrix equations.

Read rows across and columns down

A matrix is an array of entries. Its size is written rows by columns. In this qualification, the calculations use two-by-two matrices and two-by-one column vectors. A scalar is an ordinary number multiplying every entry.

For a matrix product, an entry comes from one row of the first matrix and one column of the second: multiply matching positions and add. Matrix multiplication is not entry-by-entry multiplication.

Worked example 1

Find 3A when A=\begin{pmatrix}2&-1\\0&4\end{pmatrix}

  1. Multiply each of the four entries by 3. The top-left entry becomes 6

  2. Your turn. What is the new top-right entry, 3(-1)?

    The top-right entry is -3

  3. Answer: 3A=\begin{pmatrix}6&-3\\0&12\end{pmatrix}

Worked example 2

Calculate \begin{pmatrix}2&-1\\3&4\end{pmatrix}\begin{pmatrix}5\\-2\end{pmatrix}

  1. Top entry: 2(5)+(-1)(-2)=10+2=12

  2. Your turn. Calculate the bottom entry: 3(5)+4(-2)

    15-8=7

  3. Answer: \begin{pmatrix}12\\7\end{pmatrix}. A two-by-two matrix times a two-by-one vector gives a two-by-one vector.

Explore row-by-column multiplication

For this matrix, the output is (2u-v, u+2v). The two entries of the input contribute to both output entries.

Worked example 3

Let A=\begin{pmatrix}1&2\\-3&1\end{pmatrix} and B=\begin{pmatrix}2&-1\\4&3\end{pmatrix}. Find AB

  1. First row times each column: 1(2)+2(4)=10 and 1(-1)+2(3)=5

  2. Your turn. Find the bottom-left entry: (-3)(2)+1(4)

    The bottom-left entry is -2

  3. The bottom-right entry is (-3)(-1)+1(3)=6
  4. Answer: AB=\begin{pmatrix}10&5\\-2&6\end{pmatrix}

Worked example 4

For A=\begin{pmatrix}1&2\\-3&1\end{pmatrix} and B=\begin{pmatrix}2&-1\\4&3\end{pmatrix}, find BA and compare it with AB

  1. Use rows of B this time. The first row gives 2(1)+(-1)(-3)=5 and 2(2)+(-1)(1)=3
  2. The bottom-left entry is 4(1)+3(-3)=-5

  3. Your turn. Calculate the bottom-right entry: 4(2)+3(1)

    The bottom-right entry is 11

  4. Answer: BA=\begin{pmatrix}5&3\\-5&11\end{pmatrix}, different from AB=\begin{pmatrix}10&5\\-2&6\end{pmatrix}. Order matters.

The identity matrix and matrix powers

The identity matrix is:

I=\begin{pmatrix}1&0\\0&1\end{pmatrix}

For any two-by-two matrix A:

AI=IA=A

It plays the role of 1 in multiplication.

A^2 means AA, not 2A and not squaring each entry. Calculate each entry by the row-by-column rule.

Worked example 5

For A=\begin{pmatrix}1&3\\0&1\end{pmatrix}, calculate A^2 and explain why it differs from 2A

  1. Your turn. In AA, what is the top-left entry 1(1)+3(0)?

    The top-left entry is 1

  2. The other entries are 1(3)+3(1)=6, 0(1)+1(0)=0 and 0(3)+1(1)=1
  3. Answer: A^2=\begin{pmatrix}1&6\\0&1\end{pmatrix}, whereas 2A=\begin{pmatrix}2&6\\0&2\end{pmatrix}. Matrix powers and scalar multiples are different operations.

Worked example 6

Find x and y if \begin{pmatrix}x&1\\2&y\end{pmatrix}\begin{pmatrix}3\\-2\end{pmatrix}=\begin{pmatrix}10\\-4\end{pmatrix}

  1. The row products give two equations: 3x-2=10 and 6-2y=-4

  2. Your turn. Solve 3x-2=10

    x=4

  3. From 6-2y=-4, subtract 6 to get -2y=-10, so y=5
  4. Answer: x=4, y=5. Check both entries of the output column.

Common mistakes

  • Multiplying matching positions. Use a row of the first matrix and a column of the second.
  • Reversing the matrix order. Keep the stated order: AB need not equal BA.
  • Losing a sign in a product. Work out each signed product before adding the row–column total.
  • Misreading a matrix power. A^2 means A\times A, not 2A or squared individual entries.
  • Scaling only some entries. A scalar multiplies every entry of the matrix.

Now try it: Matrix Multiplication practice questions

More on this topic: Matrix Multiplication worksheet with full solutions