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Square Matrices - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
When we are learning about matrices and matrix operations, one of the first things we need to learn about are square matrices. Square matrices have many applications in the real world. Square matrices can be used to represent and solve systems of equations, can be invertible and have determinants. The determinants of square matrices can be used to find areas and orthogonal vectors.
Let's review matrices really quickly. I have two matrices here a and b. Matrix a has 2 rows and 3 columns, matrix b has 2 columns and 3 rows. Always remember that the number of rows is the number of horizontal rows columns are vertical so we would say that the dimensions of matrix a is the number of rows times the number of columns so 2 by 3 and the dimensions of b 3 by 2. It's really important to able to recognize that the dimensions of matrix because that's what determines whether or not you can multiply 2 matrices so let's look at a problem here.
I want to multiply a and b and then b and a, here again this is matrix a and it's a 2 by 3, let me write that down over here 2 by 3, and matrix b is a 3 by 2. You can only multiply two matrices when these two inner numbers are the same so these must be the same and the result of the multiplication will have dimensions equal to this 2 times 2, so we're going to get 2 by 2 matrix out of this multiplication that's important to know. Alright, let's do the multiplication, so remember when you multiply you multiply across and down so it's going to be -2+16+18 that's going to be one entry -2+16+18 and then -1-2+15 so what I'm doing is each time I'm multiplying right? Multiplying -1 by 2, 2 by 8, 3 by 6 so I'll keep going I get 4 by 2 8 plus 0 plus 30, 8+30, and then for this entry, 4+0+25, 4+25 and so here is my 2 by 2 as predicted and I'm going to have this is 16+16 32 this is -3+15 12, 38 and 29 okay this my product.
Now let's do the reverse with we're commuting a and b let's multiply them this way notice that this is now a 3 by 2 and this is a 2 by 3 so a 3 by 2 times a 2 by 3 first of all observe we can multiply because the inner two numbers are the same and the result of the product is going to be a 3 by 3 matrix. Alright, so let's see that, so first I'm going to go across and down so -2+4 that's my first entry, 4+0 4, 6+5 now I go second row -8-4, 16+0, 24-5, third row, and notice by the way one of things you know is when you're multiplying you always multiply for the left matrix you always pick a row and for the right matrix you always pick a column and what's important about recognizing this is that the row number of the left matrix and the column number of the right matrix tells you where the entry is going to go row 3 column 1 that's right here row 3 column 1, so -6+20, 12+0, 18+25 okay so we're going to do a bunch of addition. This is going to be 2 let's just go down the column -12 and 14; 4, 16, 12; 11, 19 and 18+25 is 43 there it is that's our 3 by 3 product, so remember you can only multiply two matrices if the number of columns equals the number of rows. The number of columns of the left matrix equals the number of rows in the right matrix and the product will have the number of rows as the left matrix and the number of columns as the right matrix a 3 by 3 is what we got here.
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