Unit 2: Introduction to Linear Algebra: Determinants and Matrix

1. Matrix and Types of Matrix

Definition: A matrix is a rectangular array of numbers (or functions) arranged in m rows and n columns. It is an m × n ("m by n") matrix.

Types of Matrix:

2. Matrix Operations

A. Addition and Subtraction

You can only add or subtract matrices that have the same dimensions (same number of rows and columns). The operation is performed element-wise.

B. Scalar Multiplication

To multiply a matrix by a scalar (a single number), you multiply every element in the matrix by that scalar.

C. Matrix Multiplication (A × B)

This is the most complex operation. To multiply matrix A (dimensions m × n) by matrix B (dimensions n × p), two conditions must be met:

1. The number of columns in A (n) must equal the number of rows in B (n).
2. The resulting matrix C will have dimensions m × p.

Each element c_ij in the new matrix is found by taking the dot product of the i-th row of A and the j-th column of B.

3. Determinants and its properties

The determinant is a special scalar value that can be computed from a square matrix. It is denoted as det(A) or |A|.

Calculating Determinants:

• For a 2×2 Matrix:
  A = [ a b ]
      [ c d ]
  |A| = ad - bc
• For a 3×3 Matrix (Sarrus's Rule or Cofactor Expansion):

  Cofactor expansion is the general method. For any row i or column j, the determinant is the sum of each element multiplied by its cofactor.

  |A| = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃

  Where the cofactor C_ij = (-1)^i+j * M_ij (M_ij is the "minor", the determinant of the 2x2 matrix left after removing row i and column j).

Properties of Determinants:

• If you swap two rows (or columns), the sign of the determinant flips.
• If a matrix has a row (or column) of all zeros, |A| = 0.
• If a matrix has two identical rows (or columns), |A| = 0.
• |A^T| = |A| (Transpose has the same determinant).
• |A × B| = |A| × |B|

4. Transpose of a matrix

Definition: The transpose of a matrix A, denoted A^T (or A'), is the matrix obtained by swapping its rows and columns.
If A is m × n, then A^T is n × m.

Example:

Properties of Transpose:

• (A^T)^T = A
• (A + B)^T = A^T + B^T
• (k * A)^T = k * A^T (where k is a scalar)
• (A × B)^T = B^T × A^T (Note the reversal of order!)

5. Scalar products, norms, orthogonality (Vectors)

A vector is a matrix with only one row (row vector) or one column (column vector). We'll consider two vectors, u and v.

Let u = (u₁, u₂, ..., u_n) and v = (v₁, v₂, ..., v_n)

• Scalar (Dot) Product: The dot product u · v is a single scalar value. u · v = u₁v₁ + u₂v₂ + ... + u_nv_n
• Norm (Length or Magnitude): The norm of a vector u, written ||u||, is its length. ||u|| = √(u₁² + u₂² + ... + u_n²)

  Note: ||u||² = u · u

• Orthogonality: Two vectors u and v are orthogonal (perpendicular) if their dot product is zero. u · v = 0

6. Linear transformations

A transformation T is a function that maps a vector from one space to another (e.g., T(v) = w).

A transformation T is linear if it satisfies two properties for all vectors u, v and any scalar c:

1. Additivity: T(u + v) = T(u) + T(v)
2. Homogeneity: T(cu) = cT(u)

Matrix Representation: Every linear transformation T can be represented by matrix multiplication. T(x) = Ax, where A is the "transformation matrix".

Elementary Operations: These are operations on the rows of a matrix (used in Gaussian elimination to solve systems of equations).

1. Swapping two rows.
2. Multiplying a row by a non-zero scalar.
3. Adding a multiple of one row to another row.

7. Solution of simultaneous linear equations

A system of n linear equations with n variables can be written in matrix form as:

• A is the n × n coefficient matrix.
• X is the n × 1 vector of variables.
• B is the n × 1 vector of constants.

Method 1: Matrix Inverse Method

If the coefficient matrix A is non-singular (i.e., |A| ≠ 0), it has an inverse, A^-1.

1. Start with AX = B
2. Multiply both sides by A^-1 (on the left): A^-1(AX) = A^-1B
3. Since A^-1A = I (Identity matrix): (A^-1A)X = A^-1B => IX = A^-1B
4. The solution is: X = A^-1 B

Method 2: Cramer's Rule

This rule gives a direct formula for each variable in the solution vector X, but is only practical for 2x2 or 3x3 systems.
The solution for a variable x_i is:

x_i = |A_i| / |A|

• |A| is the determinant of the main coefficient matrix A.
• |A_i| is the determinant of a special matrix (A_i) formed by replacing the i-th column of A with the constant vector B.

8. Economic applications of matrix algebra

A. Solving Market Equilibrium

Consider a 2-good market (e.g., apples and bananas). The demand and supply for each good depends on the price of both goods. We can set Q_d = Q_s for both markets to get a system of linear equations in the prices P_a and P_b. We can then use Cramer's Rule or the Inverse Method to find the equilibrium prices.

B. Leontief Input-Output Model

This is a major application of linear algebra in economics, showing how the output of one industry is an input for another.

• X = Total Output vector (What each industry produces).
• D = Final Demand vector (What consumers want).
• A = Technology Matrix (or Input-Output Coefficient matrix). An element a_ij shows how much input from industry i is needed to produce 1 unit of output for industry j.

The fundamental equation is:
Total Output = Intermediate Demand + Final Demand
X = AX + D

To find the Total Output (X) needed to satisfy a given Final Demand (D):

1. X - AX = D
2. (I - A)X = D (where I is the identity matrix)
3. X = (I - A)^-1 D

The matrix (I - A)^-1 is called the Leontief Inverse Matrix. Each element tells you the total output from industry i required to deliver 1 unit of final output to industry j, accounting for all knock-on effects.