Multiplying matrices by scalars

now that we know what a matrix is let's see if we can start to define some operations on matrices so let's say I have the 2x3 matrix so 2 rows and 3 columns and the entries are 7 5 negative 10 3 8 and 0 and I want to define what happens when I multiply 3 times this whole thing so first of all let's get a little terminology out of the way the number 3 in just the everyday world if you weren't dealing with matrices or vectors and if you don't know what vectors are don't worry about them just now you would just call that a number you would call this a real number it's just a regular number sitting out there but now in the world where we have these new structured things these matrices these these arrays of numbers we will refer to these just plain old real numbers that aren't part of some type of an array here we call these scalars we call this we call this a scalar so essentially what we're defining here we don't know we have I haven't said what this is actually going to turn out to be but whatever this turns out to be will be a product of scalar multiplication or we're multiplying a scalar times a matrix and so how would you define this what do you think this should be 3 times this stuff right over here well the world could have defined scalar multiplication however it saw fit but one way that we find perhaps most obvious and the most useful is to multiply this scalar quantity times each of the entries so this is going to be equal to 3 times 7 in the top left 3 times 5 3 times negative 10 3 times 3 3 times 8 and 3 times 0 which will give us it didn't change the dimensions of the matrix it didn't change I guess you could say the structure of the matrix it just multiplied each of the entries times 3 so the top left entry is now going to be 21 the middle entry or the the middle the entry in the middle row top column is going to be 15 negative 39 24 and 0 so when you multiply a matrix times a scalar you just multiply each of those entries times that scalar quantity