Multiplying matrices
Matrix multiplication uses rows from the first matrix and columns from the second matrix. Each product entry is the sum of the products of matching positions
\[(AB)_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots\]
- Check the dimensions The number of columns in the first matrix must equal the number of rows in the second matrix \(\displaystyle A_{2 \times 2}B_{2 \times 2}\text{ is defined}\)
- Use a row and a column Each product entry uses one row from the first matrix and one column from the second matrix \(\displaystyle (AB)_{11}=\begin{bmatrix}2&3\end{bmatrix}\begin{bmatrix}5\\2\end{bmatrix}\)
- Multiply matching positions Pair the first row entry with the first column entry, then continue in order \(\displaystyle (AB)_{11}=2 \times 5+3 \times 2\)
- Add every product The paired products are added to give one entry of the product matrix \(\displaystyle 2 \times 5+3 \times 2=10+6=16\)
Worked example
Find AB for the two matrices shown below
\[A=\begin{bmatrix}2&3\\1&4\end{bmatrix},\quad B=\begin{bmatrix}5&1\\2&6\end{bmatrix}\]
- \(\displaystyle (AB)_{11}=\begin{bmatrix}2&3\end{bmatrix}\begin{bmatrix}5\\2\end{bmatrix}\)Use row 1 of A with column 1 of B.
- \(\displaystyle (AB)_{11}=2 \times 5+3 \times 2=10+6=16\)Multiply the paired entries and add the two products
- \(\displaystyle (AB)_{12}=2 \times 1+3 \times 6=20,\quad (AB)_{21}=1 \times 5+4 \times 2=13\)Use row 1 with column 2, then row 2 with column 1.
- \(\displaystyle (AB)_{22}=1 \times 1+4 \times 6=25,\quad AB=\begin{bmatrix}16&20\\13&25\end{bmatrix}\)The final entry completes the product matrix
Common mistakes
Multiplying corresponding positions
✕\(\displaystyle NP=\begin{bmatrix}8&5\\3&12\end{bmatrix}\)✓\(\displaystyle NP=\begin{bmatrix}11&14\\13&17\end{bmatrix}\)
Matrix multiplication uses a complete row and column for each entry, then adds the paired productsLeaving the products unsummed
✕\(\displaystyle (AB)_{11}=2 \times 5=10\)✓\(\displaystyle (AB)_{11}=2 \times 5+3 \times 2=16\)
Every matching pair contributes a product, and all of those products must be added for one entrySwapping the row and column
✕\(\displaystyle (AB)_{11}=2 \times 5+1 \times 1=11\)✓\(\displaystyle (AB)_{11}=2 \times 5+3 \times 2=16\)
The first entry uses row 1 of A and column 1 of B, not column 1 of A and row 1 of B.Difficulty levels
introMultiply two small square matrices with non-negative entries
standardMultiply a non-square pair with signed entries and find a named entry
challengeMultiply a row by a column and decide whether reversing the factors gives the same result
examSelect a product entry from entrywise, transposed and unsummed distractors
Questions about this topic
How is one entry of a matrix product found?+
Use the matching row of the first matrix and column of the second matrix. Multiply corresponding entries and add all the products
What order does a matrix product have?+
The product has the number of rows of the first matrix and the number of columns of the second matrix, provided the product is defined
Why are the products added?+
Each product represents one matching pair in the row and column. Their sum is the dot product that gives one entry of the matrix product
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices
In this chapter
State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals