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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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 $m×n$ and a $n×p$ matrix is an $m×p$ matrix.

$⎣⎢⎢⎡ a_{11}⋮a_{m1} ⋯⋱⋯ a_{1n}⋮a_{mn} ⎦⎥⎥⎤ ⋅⎣⎢⎢⎡ b_{11}⋮b_{n1} ⋯⋱⋯ b_{1p}⋮b_{np} ⎦⎥⎥⎤ =⎣⎢⎢⎡ c_{11}⋮c_{m1} ⋯⋱⋯ c_{1p}⋮c_{mp} ⎦⎥⎥⎤ $

- The element in the $ith$ row and $jth$ column of the product matrix is calculated using the elements in the $ith$ row of the first matrix and $jth$ column of the second matrix using the following formula.

$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+⋯+a_{in}b_{nj} $ Click on the elements of the product matrix below to see the illustration of how to find the product of a $4×2$ and a $2×3$ matrix.

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