Matrix Operations
The name of the process of turning any matrix into row echelon form.
Row reduction OR Gaussian Elimination
[3 -1] x [-2]
[4 2] [ 5]
[-11]
[ 2 ]
A system of linear equations with 3 variables will have this many solutions to find.
3
[5 4] - [2 -3]
[3 7]
A matrix with "M" rows and "N" columns:
The numbers in a matrix at (1,1), (2,2), and (3,3) are along this.
The main diagonal
-x -2y = 4
2x +8y = -28
Find the solution.
(6, -5)
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[2 3 5] + [3 5 8]
[5 8 13]
A= [7 3] B= [9 6] C= [0 -11]
[-4 -1] [3 10] [2 8]
Find 2C+A
[7 -19]
[0 15]
Other than multiplication/division of a single row, and add/subtraction of rows, what is the third matrix operation we use in Gaussian Elimination?
-x -8y = 49
-x -2y = 7
Find the solution.
(7, -7)
x+3y+9z=4
-x+8y-6z=4
x+3y+3z=10
Find the solution
x=10, y=1, z=-1
(10, 1, -1)
[1 -2 -1] [-2 1 2]
[-3 4 3] - [4 3 -4]
[5 -6 -5] [-6 -5 6]
[3 -3 -3]
[-7 1 7]
[11 -1 -11]
Row by row this is how we go... the process for which operation... do you know?
Matrix Multiplication
What is the rule for determining if 2 matrices can be added together?
They have to be the exact same dimensions (equal number of columns).
-2x + y = 0
10x + 7y = 48
Find the solution.
(2,4)
-9x+6y+4z=-10
-9x+5y+4z=-8
9x+y-3z=1
Find the solution.
x=2, y=-2, z=5
(2, -2, 5)
[-2 6] + [-3 11]
[1 4] [2 7]
[-5 17]
[3 11]
[1 -2] [-7 8]
2 x [-3 4] + [9 -10]
[5 -6] [-11 12]
[-5 4]
[3 -2]
[-1 0]
The minimum number of zeros in a 4x4 matrix that is in row echelon form.
6
9x + 2y = -38
9x + 10y = 26
Find the solution.
(-6,8)
7x + y + 5z = 27
4x + 3y + 5z = 21
6x + y +2z = 9
Find the solution.
x=0, y=-3, z=6
(0, -3, 6)
X= [2 -3] Y= [-4 1]
[-1 4] [3 -2]
Find X-Y
[-6 4]
[4 -6]
Multiply the following matrices:
[1 2] [1 4]
[3 6] x [5 2]
[5 6] [3 6]
Impossible
(3x2 and 3x2)