- To computea givenelementof the productmatrix,we multiplyeachelementof that row in the left matrix
by the correspondingelementin that columnin the right matrix,and add theseproducts.
Someof this informationcan be statedeasilyin symbols.
If is an -by- matrix,then mustbe an -by- matrixin orderto find the product
.
is an -by- matrix.Let’s look againat matrices and above:
We found. Is true?Surprisingly, we cannotevencalculate. This wouldhaveus
multiplyinga left matrixthat is -by- timesa right matrixthat is -by-. This doesnot satisfythe re-
quirementsstatedabove.It’s not that doesnotequal - the fact is, doesnot evenexist!
Conclusion:Multiplicationof matricesis not commutative.
Translatedloosely, somematricesyou can’tevenmultiply, and for somematricesthat you can multiply, the
operationis not commutative.
Example 3
Do the followingoperation:Notice:This multiplicationin effect addsthe elementsof eachrow of the left matrixfor the first elementin
the productmatrix,and insertsa for the secondelementin eachrow of the productmatrix.
MatrixMultiplicationand Reflections
We knowfrom earlierworkhow reflectionsin the -axis,the -axis,and the line affect the coor-
dinatesof a point.Thoseresultsare summarizedin the followingdiagram.