(1 2) (5 6)
(3 4) (7 8)
The way we multiply matrices is column by column. To find the first column in the product, we multiply the matrix A by the first column of B, and that’s the answer. To find the second column of the product, we multiply A by the second column of B.
Step 1: Finding the first column of the product:
(1 2) (5) (1*5 + 2*7) (19)
(3 4) (7) = (3*5 + 4*7) = (43)
Step 2: Finding the second column of the product:
(1 2) (6) (1*6 + 2*8) (22)
(3 4) (8) = (3*6 + 4*8) = (50)
Step 3: Combining the above:
(1 2) (5 6) (19 22)
(3 4) (7 8) = (43 50)
Let’s do a harder one: Let the matrices C and D be (respectively)
(1 2 3) (3 0)
(4 5 6) and (1 2)
(2 1 0) (0 5)
First, let’s check and make sure we can multiply CD. C is 3x3, D is 3x2, so yes we can, and the product will be 3x2.
Step 1: C times the first column of D gives the first column of CD
(1 2 3) (3) (1*3 + 2*1 + 3*0) ( 5)
(4 5 6) (1) = (4*3 + 5*1 + 6*0) = (17)
(2 1 0) (0) (2*3 + 1*1 + 0*0) ( 7)
Step 2: C times the second column of D gives the second column of CD
(1 2 3) (0) (1*0 + 2*2 + 3*5) (19)
(4 5 6) (2) = (4*0 + 5*2 + 6*5) = (40)
(2 1 0) (5) (2*0 + 1*2 + 0*5) ( 2)
Step 3: Combining the above yields CD =
(1 2 3) (3 0) (5 19)
(4 5 6) (1 2) = (17 40)
(2 1 0) (0 5) ( 7 2 )
[2] GAUSSIAN ELIMINATION:
Matrices can be used to represent systems of equations, which we then try to solve. For example, let’s say we have the two equations:
3x + 2y = 5
4x + 5y = 7
Then we can write this in matrix form by
(3 2) (x) (5)
(4 5) (y) = (7)
Or, if we had the three equations
3x + 2y + 5z = 8
2x + 2y + 4z = 7
7x + 9y + 0z = 1
Then we can write this in matrix form by
(3 2 5) (x) (8)
(2 2 4) (y) = (7)
(7 9 0) (z) (1)
Now, we want to find a way to solve such systems of equations. Let’s start with an easy example:
1x + 2y = 1
3x + 7y = 2
We can write this in matrix form by
(1 2) (x) (1)
(3 7) (y) = (2)
Now, let’s look at the two equations. If we multiply the first equation by -3 we get: -3x -6y = -3. If we then add this to the second equation (3x + 7y = 2) we get a new second equation:
3x + 7y = 2
+ -3x - 6y = -3
------------------
0x + 1y = -1
So now we have the two equations
1x + 2y = 1
0x + 1y = -1
which we can write in matrix form as
(1 2) (x) ( 1)
(0 1) (y) = (-1)
We started with the matrix
(1 2) (x) (1)
(3 7) (y) = (2)
If we multiply the first row by -3 and add that to the second row, we get the matrix
(1 2)
(0 1)
And if we multiply 1 by -3 and add it to 2 we get the vector
( 1)
(-1)
So we see we can symbolically represent multiplying and adding equations by multiplying and adding rows. Slowly, here goes:
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