Matrices and Matrix Operations

A Simple Economic Model

\(q\): quantity of hats, \(p\): price of a single hat
Demand for hats: \(q = 100-2p\)

A Simple Economic Model

\(q\): quantity of hats, \(p\): price of a single hat
Supply for hats: \(q = 20+3p\)

A Simple Economic Model

\(q\): quantity of hats, \(p\): price of a single hat
 
Demand for hats: \[q = 100-2p\] Supply for hats: \[q = 20+3p\]
 
Equilibrium: At what price will both demand and supply be equal?

Equilibrium

Equilibrium: At what price will both demand and supply be equal? \[100-2p = 20+3p \rightarrow p^* = \$16\]
 
What is the quantity traded at this price? \[q ^* = 100-2 \times 16 = 20+3 \times 16 = 68\]
 
\(q ^*\) and \(p^*\) are determined simultaneously.

Equilibrium

Matrix Algebra

We solved a system of two (linear) equations in two variables.
 
Complex economic models: multiple equations with multiple variables
 
Hard to just wing it... Enter, Matrix Algebra!
 
Matrix Algebra can help us write complex system of equations compactly and solve them

A Simple Economic Model

\(q\): quantity of hats, \(p\): price of a single hat
 
Demand for hats: \(q = 100-2p\)
 
Supply for hats: \(q = 20+3p\)
 
Rewrite the two equations: \[\begin{aligned} q + 2p &= 100 \\ q-3p &= 20 \end{aligned}\]

A Simple Economic Model

Two equations in two unknowns: \[\begin{aligned} q + 2p &= 100 \\ q-3p &= 20 \end{aligned}\]

Can write this as \(Ax = b\), where \[A = \begin{bmatrix} 1 & 2 \\ 1 & -3 \end{bmatrix} \quad x = \begin{bmatrix} q \\ p \end{bmatrix} \quad b = \begin{bmatrix} 100 \\ 20 \end{bmatrix}\]

These arrays are called matrices.

Matrices

A matrix is a rectangular array of numbers, parameters, or vectors.

Example. \(A = \begin{bmatrix} 2 & 3 & 1 \\ -1 &4 & 6 \end{bmatrix}\)

Dimensions of matrix:

• Number of rows (\(m\))

• Number of columns (\(n\))

Dimension of a Matrix

A matrix with \(m\) rows and \(n\) columns is referred to as an \(m \times n\) matrix
 
What’s the dimension of \(A\)?
 
\[A = \begin{bmatrix} 2 & 3 & 1 \\ -1 &4 & 6 \end{bmatrix}_{\_\_ \times \_\_}\]

General Notation

\[A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & \cdots & a_{mn} \\ \end{bmatrix}\]

Can write it more compactly \[A = [ a_{ij} ] \quad i=1,2,...,m; j=1,2,...,n\]

Square Matrices

Square matrix: equal number of rows and columns
 
Example. \[A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}_{3\times 3}\]

Equal Matrices

Two matrices are equal if all their elements are identical.
 
Example. \[A = \begin{bmatrix} 1 & 8 \\ 4 &-1 \\ \end{bmatrix} \neq \begin{bmatrix} 1 & 8 \\ 4 & 2 \\ \end{bmatrix}\]

So \(A=B\) if and only if \(a_{ij} = b_{ij}\) for all \(i, j\)

Matrix Addition and Subtraction

How to add or take the difference between two matrices?

\(\rightarrow\) Element-by-element

\(\rightarrow\) Matrices have to have same dimension
 

Example. \[A = \begin{bmatrix} 2 & 3 \\ 4 & -6 \end{bmatrix} \quad \quad B = \begin{bmatrix} 1 & 8 \\ -2 & 3 \end{bmatrix}\]

What is \(A+B\) and \(A-B\)?

Scalar Multiplication

How to multiply a scalar to a matrix? \[\lambda \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ \end{bmatrix} = \begin{bmatrix} \lambda a_{11} & \lambda a_{12} & \lambda a_{13} \\ \lambda a_{21} & \lambda a_{22} & \lambda a_{23} \\ \end{bmatrix}\] Example. \[A = \begin{bmatrix} 2 & 3 \\ 4 & -6 \end{bmatrix} \quad \quad B = \begin{bmatrix} 1 & 8 \\ -2 & 3 \end{bmatrix}\] What is \(2B\) and \(A-2B\)?

Matrix Multiplication

A whole new animal...

Only possible to multiply two matrices, \(A_{m \times n}\) and \(B_{p \times q}\) to get \(AB\) if \(n = p\) i.e. \[\text{number of columns in } A = \text{ number of rows in } B\]

Example. \(A = \begin{bmatrix} 2 & 3 & 1 \\ 4 & -6 & -2 \end{bmatrix}_{2 \times 3}\) \(B = \begin{bmatrix} 1 & 8 \\ -2 & 3 \end{bmatrix}_{2 \times 2}\)

Cannot do \(AB\), but can do \(BA\)

Matrix Multiplication

Another example. \[A = \begin{bmatrix} 2 & 3 & 1 \\ 4 & -6 & -2 \end{bmatrix}_{2 \times 3} B = \begin{bmatrix} 1 \\ -2 \\ 4 \end{bmatrix}_{3 \times 1}\]

Can we multiply \(A\) and \(B\) to find \(C=AB\)?

Yes, since \(A\) has 3 columns, which is equal to the number of rows in \(B\).

Also, the dimension of \(C\) will be \(2 \times 1\).

Matrix Multiplication

So how to actually multiply these matrices?

\[C = AB\] \[c_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + ... + a_{in} b_{nj} = \sum_{k=1}^n a_{ik} b_{kj}\]

The element \(c_{ij}\) is obtained by multiplying term-by-term the entries of the \(i\)th row of \(A\) and \(j\)th column of \(B\).

Matrix Multiplication: Examples

\[A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ \end{bmatrix}_{2 \times 3} B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \\ \end{bmatrix}_{3 \times 2}\]

Here, \[C = AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} \\ a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32} \\ \end{bmatrix}_{2 \times 2}\]

Homework Questions

• Exercise 4.2: 1, 2, 4

Textbook reference: 4.1, 4.2