Welcome to your ultimate and comprehensive study guide for Chapter 1: Matrices and Determinants. Matrices and determinants are widely used in mathematics, physics, statistics, electronics, and computer science. This chapter explores matrix definitions, rows and columns, order of matrices, types of matrices (such as square, rectangular, identity, symmetric, and skew-symmetric), matrix addition and multiplication, determinants, singular and non-singular matrices, inverses, and solving linear systems using matrix inversion and Cramer’s rule.
1. Introduction to Matrices and Basic Terminology
A rectangular array or a formation of a collection of real numbers enclosed by square brackets [] is said to form a matrix.
- Rows and Columns: Horizontal entries form the rows of a matrix, while vertical entries form its columns.
- Order of a Matrix: If a matrix M has m rows and n columns, its order is specified as m-by-n.
- Equal Matrices: Two matrices A and B are equal (A = B) if and only if they have the same order and their corresponding entries are equal.
2. Types of Matrices
- Row and Column Matrices: A matrix with only one row is a row matrix, and a matrix with only one column is a column matrix.
- Rectangular and Square Matrices: A matrix where the number of rows is not equal to columns is rectangular; if rows equal columns, it is a square matrix.
- Null (Zero) Matrix: A matrix where each of its entries is zero.
- Symmetric and Skew-Symmetric: A square matrix A is symmetric if At = A, and skew-symmetric if At = -A.
- Diagonal, Scalar, and Identity Matrices: A square matrix with non-zero diagonal entries and zero non-diagonal entries is a diagonal matrix. If diagonal entries are equal, it is scalar; if all diagonal entries are 1, it is an identity (unit) matrix.
3. Matrix Operations: Addition, Subtraction, and Multiplication
- Addition and Subtraction: Two matrices of the same order can be added or subtracted by adding or subtracting their corresponding entries. Matrix addition satisfies commutative and associative laws.
- Scalar Multiplication: Multiplying a matrix by a real number (scalar) k multiplies every entry of the matrix by k.
- Multiplication of Matrices: Two matrices A and B are conformable for multiplication (AB) if the number of columns of A equals the number of rows of B. Note that commutative law of multiplication does not hold in general (AB ≠ BA).
4. Determinants and Multiplicative Inverse
- Determinant of a 2-by-2 Matrix: For a square matrix M = [[a, b], [c, d]], the determinant is given by det M = ad – bc.
- Singular and Non-Singular: A matrix is singular if its determinant is zero (|A| = 0), and non-singular if its determinant is non-zero (|A| ≠ 0).
- Adjoint and Inverse: The inverse of a non-singular matrix A is given by A-1 = (Adj A) / |A|, satisfying AA-1 = I = A-1A.
5. Step-by-Step Example: Finding the Inverse of a Matrix
Example: Find the multiplicative inverse of matrix A = [[-1, 3], [2, 0]].
Step-by-Step Solution:
- Step 1: Compute the determinant of A:
|A| = (-1)(0) – (3)(2) = 0 – 6 = -6. Since |A| ≠ 0, the inverse exists. - Step 2: Find the adjoint of A: Interchange diagonal elements and change signs of off-diagonal elements.
Adj A = [[0, -3], [-2, -1]]. - Step 3: Apply the inverse formula (A-1 = Adj A / |A|):
A-1 = (1 / -6) * [[0, -3], [-2, -1]] = [[0, 1/2], [1/3, 1/6]].
Essential Conceptual Review Questions
Q1: When is a matrix considered conformable for multiplication?
Answer: Two matrices A and B can be multiplied to produce AB only if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B).
Q2: What is the primary condition for a matrix to possess a multiplicative inverse?
Answer: A matrix must be a non-square matrix? No—it must be a non-singular square matrix, meaning its determinant must not equal zero (|A| ≠ 0).