Introduction to Matrices

Table of Contents
1. What is a Matrix?
2. Order of a Matrix
3. Transpose of a Matrix
4. Singular and Nonsingular Matrices
5. Matrix Addition and Multiplication: Properties
6. Types of Matrices
7. Adjoint (Classical Adjoint) of a Square Matrix
8. Inverse of a Square Matrix
9. Trace of a Matrix
10. Additional Important Concepts (brief)
11. Solved Examples
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What is a Matrix?

Definition: A matrix is a rectangular array of numbers (or symbols) arranged in rows and columns. Elements of a matrix are enclosed in parentheses or square brackets and are referred to by their row and column indices.

For example, the matrix below has nine elements and is of order 3 × 3.

Each element of the matrix M can be denoted by a_ij, which means the element in the i-th row and j-th column. For example, a₂₃ = 6 in the matrix above.

Order of a Matrix

Order (or dimension): The order of a matrix is given by the number of rows and the number of columns and is written as rows × columns.

Thus the sample matrix M above is of order 3 × 3.

Transpose of a Matrix

Definition: The transpose of an m × n matrix A, denoted by A^T, is the n × m matrix obtained by interchanging rows and columns of A.

If A = [a_ij] is m × n, then A^T = [b_ij] is n × m where b_ij = a_ji.

Properties of Transpose

• (A^T)^T = A
• (A + B)^T = A^T + B^T
• (AB)^T = B^TA^T
• (cA)^T = cA^T for any scalar c

Singular and Nonsingular Matrices

• Singular matrix: A square matrix A is singular if its determinant is zero; that is, |A| = 0. A singular matrix is not invertible.
• Nonsingular (invertible) matrix: A square matrix A is nonsingular if its determinant is non-zero; that is, |A| ≠ 0. A nonsingular matrix has an inverse.

Matrix Addition and Multiplication: Properties

• Commutative property of addition: For any two matrices A and B of the same order, A + B = B + A.
• Associative property of addition: For A, B, C of the same order, (A + B) + C = A + (B + C).
• Non-commutative property of multiplication: In general, matrix multiplication is not commutative; for most A and B, AB ≠ BA.
• Associative property of multiplication: For conformable matrices, (AB)C = A(BC).
• Distributive property: Matrix multiplication distributes over addition: A(B + C) = AB + AC and (A + B)C = AC + BC.
• Compatibility for multiplication: If A is of order m × n and B is of order n × p, then the product AB is defined and has order m × p.

Types of Matrices

• Square matrix: A matrix with the same number of rows and columns (n × n).
• Symmetric matrix: A square matrix A is symmetric if A^T = A. Elements are mirrored about the main diagonal.
• Skew-symmetric (or antisymmetric) matrix: A square matrix A is skew-symmetric if A^T = -A. Diagonal entries of a real skew-symmetric matrix are zero.
• Diagonal matrix: A square matrix whose off-diagonal entries are zero. Only diagonal entries may be non-zero.
• Identity matrix: Denoted I_n, it is an n × n diagonal matrix with ones on the main diagonal and zeros elsewhere. For any conformable A, AI = IA = A.
• Orthogonal matrix: A square matrix Q with real entries is orthogonal if Q Q^T = Q^T Q = I. Columns (and rows) of Q form an orthonormal set.
• Idempotent matrix: A matrix A is idempotent if A² = A.
• Involutory matrix: A matrix A is involutory if A² = I, i.e. A is its own inverse.

Note: Every square matrix A can be uniquely decomposed as the sum of a symmetric and a skew-symmetric matrix:
A = 1/2 (A^T + A) + 1/2 (A - A^T).

Adjoint (Classical Adjoint) of a Square Matrix

Definition: The adjoint (also called the classical adjoint) of an n × n matrix A, denoted adj(A), is the transpose of the cofactor matrix of A.

Properties of Adjoint

1. A · adj(A) = adj(A) · A = |A| I_n
2. adj(AB) = adj(B) · adj(A)
3. |adj(A)| = |A|^n-1
4. adj(kA) = k^n-1 adj(A) for scalar k
5. |adj(adj(A))| = |A|^(n-1)2
6. adj(adj(A)) = |A|^n-2 A
7. If A = [L, M, N] (block notation) then adj(A) = [MN, LN, LM] (block-wise relation for 3×3 block matrices under suitable conditions)
8. adj(I) = I

Where n is the order of the square matrix A.

Inverse of a Square Matrix

Definition: For a square matrix A of order n, the inverse A^-1 (if it exists) is the matrix such that A A^-1 = A^-1 A = I.

The inverse of a Square Matrix

An inverse exists only when |A| ≠ 0; that is, A must be nonsingular.

Properties of Inverse

1. (A^-1)^-1 = A
2. (AB)^-1 = B^-1 A^-1
3. Only a non-singular square matrix has an inverse.

Where to use the Inverse Matrix

Systems of linear equations can be written in matrix form as AX = B. If A is invertible, the unique solution is X = A^-1 B. This method is particularly useful when A is fixed and several different right-hand sides B must be solved.

Example system (corrected for clarity):

7x + 2y + z = 21

3y - z = 5

-3x + 4y - 2z = -1

If the coefficient matrix is A and the variable vector is X = [x, y, z]^T, then X = A^-1 B provided |A| ≠ 0.

Trace of a Matrix

Definition: The trace of a square matrix A, denoted tr(A), is the sum of its diagonal elements.

Property: The trace of a matrix equals the sum of its eigenvalues (counted with algebraic multiplicity).

For example:

Additional Important Concepts (brief)

• Determinant: A scalar associated with a square matrix; used to test invertibility, compute areas/volumes under linear maps and in many formulae (Laplace expansion, product rule |AB| = |A||B|).
• Rank: The rank of a matrix is the maximum number of linearly independent rows (or columns). Rank determines solvability of linear systems (compare rank of augmented matrix and coefficient matrix).
• Row-reduction and Echelon forms: Elementary row operations reduce matrices to row echelon form (REF) or reduced row echelon form (RREF) for solving linear systems and finding rank.
• Linear transformations: Any m × n matrix represents a linear map from Rⁿ to R^m; properties of the matrix correspond to properties of the transformation (invertible ↔ bijective).
• Applications (engineering): Matrices are used in solving circuits (nodal and mesh methods), structural analysis (stiffness matrices), control systems, image processing, computer graphics, data fitting, and modelling networks.

Solved Examples

Q1: For matrices of the same dimension M, N, and scalar c, which one of these properties DOES NOT ALWAYS hold?
(a) (M^T)^T = M
(b) (cM)^T = c(M)^T
(c) (M + N)^T = M^T + N^T
(d) MN = NM
Ans: (d)
Sol: 

Consider two 2 × 2 matrices M and N (same dimension):

Compute M × N:

We observe that (M × N)_2×2 ≠ (N × M)_2×2, even when dimensions are equal.

However, if M and N are both the 2 × 2 identity matrix then (M × N) = (N × M) = I.

Therefore (M × N)_2×2 is not always equal to (N × M)_2×2, and option (d) does not always hold.

Q2: For A = 

 the determinant of A^TA^-1 is:
(a) sec² x
(b) cos 4x
(c) 1
(d) 0 
Ans: (c)

Sol: 

Explanation:

If A is given and invertible then |A^T A^-1| = |A^T| · |A^-1|.

Use |A^T| = |A| and |A^-1| = 1/|A| to obtain |A^T A^-1| = |A| · (1/|A|) = 1.

Q3: Let X be a square matrix. Consider the following two statements on X.
I. X is invertible.
II. Determinant of X is non-zero.
Which one of the following is TRUE?
(a) I implies II; II does not imply I.
(b) II implies I; I does not imply II.
(c) I does not imply II; II does not imply I.
(d) I and II are equivalent statements.
Ans: (d)
Sol: 

If X is invertible then X^-1 exists and therefore |X| ≠ 0.

If |X| ≠ 0 then X has an inverse (the adjoint formula A^-1 = adj(A)/|A|), so X is invertible.

Hence I implies II and II implies I; both statements are equivalent.

Q4: The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?
(a) PQ = I but QP ≠ I
(b) QP = I but PQ ≠ I
(c) PQ = I and QP = I
(d) PQ - QP = I
Ans: (c)
Sol:

Given P = Q^-1.

Post-multiply by Q:

PQ = Q^-1 Q

PQ = I (because Q^-1 Q = I)

Pre-multiply by Q:

QP = Q Q^-1

QP = I (because Q Q^-1 = I)

So, PQ = I and QP = I; option (c) is correct.