Matrices (Types, Operations, Transpose, Inverse Basics)

1) Why matrices in economics?

Matrices are used to represent and solve systems of equations in:

• market equilibrium with many goods,
• national income models,
• input–output analysis (Leontief),
• econometrics/regression (X'X, etc. at higher level),
• optimization constraints.

2) Matrix: definition and notation

A matrix is a rectangular array of numbers arranged in rows and columns.

• $m$ rows, $n$ columns
• element in i-th row, j-th column is $a_{ij}$

3) Types of matrices (high-yield)

• Row matrix: $1\times n$
• Column matrix: $m\times 1$
• Square matrix: $n\times n$
• Zero (null) matrix: all entries 0
• Diagonal matrix: square; nonzero only on main diagonal
• Identity matrix $I$: diagonal with 1s on main diagonal
• Scalar matrix: diagonal with same constant $c$ on diagonal
• Upper triangular: entries below diagonal are zero
• Lower triangular: entries above diagonal are zero
• Symmetric: $A=A'$ (transpose equals itself)
• Skew-symmetric: $A'=-A$ (diagonal must be 0)

4) Basic operations

4.1 Equality

Two matrices are equal if same order and all corresponding elements equal.

4.2 Addition and subtraction

Only possible if matrices have same order: