Math 4. Math for Economists. Lecture 05

UCI Math 4: Math for Economists (Summer 2013)
Lec 05. Math for Economists
Instructor: Jason Kronewetter, Ph.D.

License: Creative Commons CC-BY-SA

Description: UCI Math 4 covers the following topics: linear algebra and multivariable differential calculus suitable for economic applications.

Recorded on August 14, 2013

[joshuatsao]

Recap of the following lesson basically talks about cofactor expansion, adjoints, real life economic examples and Cramer's Rule.

1. Solving for determinant using Cofactor Expansion
2. Proofs
2.1 |AB| = |BA|
2.2 A•A^T=I |A| = +-1
3. Adjoint(A)
4. Inverse of nxn matrices
5. Input-Output Analysis (Economics Example)
6. Cramer's Rule
6.1 1x1
6.2 2x2
6.3 3x3

Hope this helps!

[pearlite]

00:00 Quick recap of previous lesson
|A| = "determinant of A" = cofactor expansion along any row or column
cofactor = c_ij = (-1)^(i+j) M_ij where M_ij is the minor of A (which is the determinant of a matrix that does not include the row i or the column j)

02:02 Example of finding the determinant of a 4x4 matrix
We can use the shortcut discussed at the end of the previous lesson to figure out the sign for the (-1)^(i+j) term in the cofactor
To solve faster, pick a row or column which has many 0s as that will mean there are fewer computations. It's also easier if you pick rows or columns that have small numbers in them like 1s.
09:00 Example of finding the determinant of another 4x4 matrix
In this example, row operations will be performed to simplify the matrix before solving for the determinant. As we learnt in the previous lesson, when you modify a matrix using the row operation of R_i=alpha*R_j + beta*R_i, the determinant remains the same.

18:40 Proof for: | A B | = | B A |

24:00 Proof for: A A^T = I (capital 'i', the identity matrix), and thus |A| = +/- 1
The determinant of the Identity Matrix is 1 i.e. |I| = 1

30:22 THE INVERSE OF AN n x n MATRIX
For every value in the matrix A, there is a corresponding cofactor. Thus we can write a matrix of the cofactors.
The "ADJOINT OF A", which can be written as "Adj(A)", is the cofactor matrix of A transposed. The result is related to the inverse of the matrix.
34:02 Example of finding the inverse of a 2x2 matrix
37:00 Formula for inverse of 2x2 matrix, which can be written as A^(-1) = (1/|A|) Adj(A)
37:44 This is the same formula for finding the inverse of an n x n matrix: A^(-1) = (1/|A|) Adj(A), where |A| does not equal 0
39:08 Example of finding the inverse for a 3x3 matrix

53:46 Example that uses economics: Input-Output analysis
Two industry model - Electricity (E) & Water (W)
We want output = demand
Demand is broken into two parts: 1) internal demand (from production), 2) external demand (from population), so output = internal demand + external demand

For the internal demand:
$1 of E needs $0.30 of E and $0.20 of W
$1 of E needs $0.20 of E and $0.40 of W

This information is organised in a 2x2 technology matrix, M, where the rows represent the input and the columns represent the output. The top row represents the electricity needed for a certain output, and the bottom row represents the water needed for a certain output

For the external demand:
$12 million of E
$8 million of W
Matrix for external demand, D is a column vector where d_E is the demand for electricity and d_W is the demand for water, in units of $million.

Matrix for Output, X, is a column vector where x_1 = output for electricity & x_2 = output for water

If output = demand, then this means X = Mx + D. Now we must solve for x.
X-Mx = D
X(I-M) = D (as the identity matrix, I, serves as 1 in the matrix world)
X = (1-M)^(-1) D

01:16:18 Cramer's rule
Given any system of n linear equations with n variables, we can write it as Ax = b, where A is the matrix of cofficients, x is the column vector containing the variables, and b is the column vector of constants
01:20:00 Make a new matrix from A, which is A_j = the matrix obtained from A by replacing the 'j'th column with b
01:24:00 Example with A_1 --> First column is replaced with b. To get |A_1|, the first column is then expanded. So |A_1| = b_1 c_11 + b_2 c_21 + ....+ b_n c_n1
For Ax=b x=A^(-1) b = (1/|A|)Adj(A) b = (1/|A|) [Cofactor matrix]^T b
To get x_1, you would need to solve to get the first row of values in this equation. Thus, x_1 = (1/|A|) (b_1 c_11 + b_2 c_21 + ....+ b_n c_n1) = (1/|A|) |A_1|
**So Cramer's rule states that X_j = |A_j| / |A| (as long as A^(-1) exists)**
01:32:12 Example of Cramer's rule using 1x1 matrix
01:33:30 Example of Cramer's rule using 2x2 matrix
When solving for x, the values for b go in the first column for A_j (numerator of the equation)
When solving for y, the values for b go in the second column for A_j (numerator of the equation)
01:36:44 Example of Cramer's rule using 3x3 matrix
When calculating |A_1|, the first column is replaced with b
When calculating |A_2|, the second column is replaced with b
When calculating |A_3|, the third column is replaced with b
01:43:42 It may be preferable to use Cramer's rule (rather than row reduction) in situations where there are fractions in the matrix, or if you only need to find a single solution and not all of them

[danielscott]

The teaching style is great, but the cameraman keeps cutting off the equations. With a small in tweak in camera awareness, this could be far faster to digest. Nonetheless, thanks for your intuitive teaching style Jason! I'm loving it.

[poopoo]

There you go! You guys still want to study  LOL

[FrankCirillo]

LOL at 18:16-18:29 you can see a student playing Civ in class XD

[Fortuna_]

At 41:00 to 43:00 when he writes the determinant for the 3x3 matrix, he just chooses the 2nd column (0, -1, 1) to use, but he did not do any row operations to make it (0, 0, 1) for example. Would that be easier (or possible) to do instead, or is this the most effective method? Just a bit confused on when you choose to do the row operations, as I thought you might want as many zero's as possible to begin with.

[alovernighter]

Does anyone know anything about the 6G math course he mentioned? Is it available somewhere on the internet?

[anaveragecactus]

i don’t go to this school but i enjoy the vibe

[actionmoviesofficial]

Maybe someone sees this and can help me. One theorem states that when you do a row operation in the form of a*r1+r2, the determinant stays unchanged. If you multiply a row by a constant, the determinant is also multiplied by this constant. So isn't the first theorem wrong if you take a very small number for a (0.1 for example) and a very big one for b (like 100). This is still a row operation but also approximately a multiplication by a constant, so neither of the two theorems should work on this operation, right?

[emmq]

bro they paid me $50 to film lectures for the entire day i was so zooted im surprised any of this was in focus

[albertinejunedin]

While solving for the Input-Output example (around 1:00:00), while formulating the example, why did he use X= Mx + D, and not just X = M + D?