Determinants
What symbol do we denote determinants with?
Straight vertical lines
ex: |A|
How do you know if two matrices are inverses?
When you multiply them together, their product equals an Identity matrix.
What is a Coefficient Matrix?
A Matrix made from all the numbers (coefficients) in front of the variables in a system of equations.
What is the rule for determining if two matrices can be added or subtracted together?
The matrices have to be the exact same dimensions (equal number of rows and columns)
Find the mistake in their steps to find the determinant, and then say what the correct answer should have been:
[[5,3],[-1,2]]
(5*2) + (3*-1)
(10) + (-3)
=7
They added the two products together instead of subtracting. The correct answer should be
10 - -3 = 13
Write a 4x4 identity matrix
[[1,0, 0,0],[0,1, 0,0],[0,0,1,0],[0,0,0,1]]
What would be the coefficient matrix of this system?
2x + 5y - 3z = 21
x - 8y + 4z = -13
-7x + y - z = 42
[[2,5, -3],[1, -8, 4],[-7, 1, -1]]
Can the following matrices be multiplied together? How do you know?
[[1,2],[3, 4],[5, 6]] [[1,2, 3],[4, 5, 6]]
Yes, because the first matrix is a 3x2, and the second matrix is a 2x3. Since the second matrix has the same number of rows, as the first one has columns (2), these matrices can be multiplied together.
Find the determinant:
[[6,4],[-9,3]]
54
Will the following matrix have an inverse? How do you know?
[[3, 3],[-1, -1]]
No it will not have an inverse. You know because the determinant is zero.
Set up the Coefficient matrix, the x-matrix, and the y-matrix (don't find any determinants, just set up what the matrices would look like)
5x - 2y = 18
3x - y = 7
Coefficient matrix -
[[5,-2],[3,-1]]
X-Matrix -
[[18,-2],[7,-1]]
Y-Matrix -
[[5,18],[3,7]]
Subtract:
[[1,-2, -1],[-3, 4, 3],[5, -6, -5]] - [[-2, 1, 2], [4, 3, -4], [-6, -5, 6]]
[[3,-3,-3],[-7,1,7],[11,-1,-11]]
Find all three separate determinants:
[[-4,3],[-1,2]] , [[6,4],[-2,3]], and [[-8,-5],[3,0]]
-5, 26, and 15
Find the inverse of the following matrix:
[[3, 4],[-2, -1]]
[[-1/5, -4/5],[2/5, 3/5]]
Set up the Coefficient matrix, the x-matrix, the y-matrix, and the z-matrix (don't find any determinants, just set up what the matrices would look like)
-5x + 3y + 4z = 20
x - y - z = -27
9x + 6y - 8z = -12
Coefficient -
[[-5,3, 4],[1,-1, -1],[9,6,-8]]
X -
[[20,3, 4],[-27,-1, -1],[-12,6,-8]]
Y -
[[-5,20, 4],[1,-27, -1],[9,-12,-8]]
Z -
[[-5,3, 20],[1,-1, -27],[9,6,-12]]
Solve the following operations:
2[[1,-2],[-3, 4],[5, -6]] + [[-7,8],[9, -10],[-11, 12]]
[[-5,4],[3,-2],[-1,0]]
Find the Determinant:
[[4,-2,6],[-1,2,0],[3,5,-7]]
-108
Are the following matrices inverse? How do you know?
[[12, 4, -7],[-2, -1, 5],[3, -8, 6]] and [[-12, -4, 7],[2, 1, -5],[-3, 8, -6]]
No, because when you multiply them together, it will not make an Identity matrix.
Solve the System using Cramer's Rule
4x - y = 19
5x - 5y = 20
(5, 1)
Solve the following multiplication:
[[1,2, -2, -1],[-3, -4, 4, 3]] xx [[5],[6],[7],[8]]
[[-5],[13]]