Unit-1: Matrices and Their Properties

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1. Elementary Operations and Rank of a Matrix

Elementary operations are used to simplify matrices. They help in finding the rank, inverse, and solving linear equations. There are three types of elementary operations:

• Row Operations (Interchange, Multiply, Add)
• Column Operations (Interchange, Multiply, Add)

The rank of a matrix is the number of non-zero rows in its row echelon form.

2. Inverse of a Matrix

The inverse of a matrix A is denoted as A^-1 and satisfies the equation:

A × A^-1 = I (Identity Matrix)

3. Solution of Linear Simultaneous Equations

Linear equations in matrix form: AX = B. The solution is given by:

X = A^-1B (if A is invertible)

4. Special Matrices and Their Properties

4.1 Orthogonal Matrix

A matrix A is orthogonal if A^T × A = I.

4.2 Symmetric Matrix

A matrix A is symmetric if A = A^T.

4.3 Skew-Symmetric Matrix

A matrix A is skew-symmetric if A^T = -A.

4.4 Hermitian Matrix

A matrix A is Hermitian if A^* = A, where A^* is the conjugate transpose.

4.5 Skew-Hermitian Matrix

A matrix A is skew-Hermitian if A^* = -A.

4.6 Normal Matrix

A matrix A is normal if A^*A = AA^*.

4.7 Unitary Matrix

A matrix A is unitary if A^*A = I.

5. Eigenvalues and Eigenvectors

For a matrix A, a scalar λ is an eigenvalue and a non-zero vector X is its corresponding eigenvector if:

A × X = λX

6. Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation:

p(A) = 0, where p(λ) is the characteristic polynomial of A.