2. ALGEBRA: MATRICES - PHARMD GURU

Introduction — What is a Matrix?

A matrix is a rectangular arrangement of numbers written in rows and columns. It is used in solving linear equations, representing data, handling transformations, and many applications in science and pharmacy.

A matrix with m rows and n columns is called an m × n matrix.

Basic Terms in Matrices

Order of matrix: Written as m × n (rows × columns).
Element: Each number inside the matrix (a_ij is the element in i-th row and j-th column).
Row matrix: Contains only one row.
Column matrix: Contains only one column.

Types of Matrices

Square matrix: Rows = columns (n × n).
Rectangular matrix: Rows ≠ columns.
Zero or null matrix: All elements are zero.
Diagonal matrix: All non-diagonal elements are zero.
Scalar matrix: A diagonal matrix where diagonal elements are equal.
Identity matrix (I): Diagonal elements = 1 and all others = 0.
Singular matrix: Determinant = 0.
Non-singular matrix: Determinant ≠ 0.
Symmetric matrix: A = A^T.
Skew-symmetric matrix: A = −A^T.

Operations on Matrices

1. Addition

Possible only when matrices have the same order (same number of rows and columns). Add corresponding elements.

2. Subtraction

Possible only when orders are same. Subtract corresponding elements.

3. Scalar Multiplication

Each element of the matrix is multiplied by a constant (scalar).

Matrix Multiplication

For matrices A (m × n) and B (n × p), product AB is defined and will be of order m × p.

Rule: Columns of A must match rows of B.

Each element of product matrix is calculated as:

(AB)_ij = Σ (a_ik · b_kj)

Important Points

Matrix multiplication is not commutative: AB ≠ BA (in general).
It is associative: (AB)C = A(BC).
It is distributive over addition: A(B + C) = AB + AC.

Transpose of a Matrix

Transpose of matrix A is written as A^T. It is obtained by interchanging rows and columns.

Example: Row 1 becomes Column 1, Row 2 becomes Column 2, and so on.

Properties of Transpose

(A^T)^T = A
(A + B)^T = A^T + B^T
(AB)^T = B^T A^T

Adjoint of a Matrix

The adjoint of matrix A is obtained by taking cofactors of all elements, forming cofactor matrix, and then taking its transpose.

adj(A) = transpose of cofactor matrix.

Inverse of a Matrix

For a non-singular matrix A (det(A) ≠ 0), inverse A⁻¹ exists and is given by:

A⁻¹ = adj(A) / det(A)

Important Notes

Inverse exists only if determinant is non-zero.
A · A⁻¹ = I (identity matrix).

Solution of System of Linear Equations

Matrix Method

System of equations can be written as AX = B.

If A is invertible:

X = A⁻¹ B

Example (Conceptual)

For equations:

x + y = 2
2x + 3y = 5

Write in matrix form and compute X using inverse.

Cayley–Hamilton Theorem

Every square matrix satisfies its own characteristic equation.

If characteristic equation is:

λ² − (trace A) λ + det(A) = 0

Then replacing λ with matrix A gives:

A² − (trace A) A + det(A)I = 0

Applications

Used to compute inverse of matrix without adjoint method.
Simplifies higher powers of matrices.

Application of Matrices in Pharmacokinetics

Matrices are useful in modelling drug concentration-time relationships, linear compartment models, and solving simultaneous differential equations in pharmacokinetics.

For example: first-order rate processes can be represented using matrix exponential solutions.

Quick Revision Points

Matrix → arrangement of numbers in rows and columns.
Square matrix: n × n.
Identity matrix: diagonal = 1.
Multiplication is not commutative.
Transpose swaps rows and columns.
Adjoint → transpose of cofactor matrix.
Inverse exists only if det(A) ≠ 0.
Cayley–Hamilton: matrix satisfies its own characteristic equation.

Detailed Notes:

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