Concept

Multiplication of Matrices

Matrices can be multiplied if the number of columns in the first matrix is the same as the number of rows in the second matrix.

The product of an $m \times n$ and a $n \times p$ matrix is an $m \times p$ matrix.

$⎣ ⎢ ⎢ ⎡ a_{11} ⋮ a_{m 1} \dots ⋱ \dots a_{1 n} ⋮ a_{m n} ⎦ ⎥ ⎥ ⎤ \cdot ⎣ ⎢ ⎢ ⎡ b_{11} ⋮ b_{n 1} \dots ⋱ \dots b_{1 p} ⋮ b_{n p} ⎦ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎡ c_{11} ⋮ c_{m 1} \dots ⋱ \dots c_{1 p} ⋮ c_{m p} ⎦ ⎥ ⎥ ⎤$

The element in the $i th$ row and $j th$ column of the product matrix is calculated using the elements in the $i th$ row of the first matrix and $j th$ column of the second matrix using the following formula.

$c_{i j} = a_{i 1} b_{1 j} + a_{i 2} b_{2 j} + \dots + a_{i n} b_{n j}$ Click on the elements of the product matrix below to see the illustration of how to find the product of a $4 \times 2$ and a $2 \times 3$ matrix.

When manual calculation is needed, rearranging the matrices on the paper can help remembering how to work out the elements.