Maths > Matrices and Determinants > 1.0 Introduction
Matrices and Determinants
1.0 Introduction
2.0 Algebra of Matrices
3.0 Special Matrices
3.1 (a) Symmetric Matrix:
3.2 (b) Skew Symmetric Matrix:
3.3 (c) Hermitian matrix:
3.4 (d) Skew Hermitian Matrix:
3.5 (e) Singular and Non-singular Matrices:
3.6 (f) Orthogonal Matrix :
3.7 (g) Idempotent Matrix :
3.8 (h) Unitary Matrix :
3.9 (i) Involuntary Matrix:
3.10 (j) Nilpotent Matrix:
4.0 Determinant of a square matrix
5.0 Adjoint of a square Matrix
6.0 Inverse of a Matrix
7.0 Types of Equations Homogenous & Non-Homogenous
8.0 Cramer's rule
9.0 Types of Linear Equations
1.4 Conjugate of a Matrix
3.2 (b) Skew Symmetric Matrix:
3.3 (c) Hermitian matrix:
3.4 (d) Skew Hermitian Matrix:
3.5 (e) Singular and Non-singular Matrices:
3.6 (f) Orthogonal Matrix :
3.7 (g) Idempotent Matrix :
3.8 (h) Unitary Matrix :
3.9 (i) Involuntary Matrix:
3.10 (j) Nilpotent Matrix:
If a matrix $A$ is having complex numbers as its elements,the matrix obtained from $A$ by replacing each element of $A$ by its conjugate $\overline {\left( {a \pm ib} \right)} = a \mp ib$ is called the conjugate of matrix $A$ and is denoted by ${\bar A}$.
Example:
If $A = \left[ {\begin{array}{c} {1 - 5i}&{5 + 3i}&{6 - 5i} \\ {4 + 2i}&{2 - 6i}&8 \\ 9&4&{3 + 7i} \end{array}} \right]$
Then, ${\bar A}$=$\left[ {\begin{array}{c} {1 + 5i}&{5 - 3i}&{6 + 5i} \\ {4 - 2i}&{2 + 6i}&8 \\ 9&4&{3 - 7i} \end{array}} \right]$
Properties of Conjugate of a Matrix
- $\overline {\left( {\overline A } \right)} = A$
- $\overline {\left( {A + B} \right)} = \overline A + \overline B $
- $\overline {\left( {\alpha A} \right)} = \overline \alpha \overline A $,$\alpha $ being any number real or complex
- $\overline {\left( {AB} \right)} = \overline A .\overline B $,where A and B being conformable for multiplication
Conjugate Transpose of a Matrix
The conjugate of the transpose of a matix $A$ is called the conjugate transpose of $A$ and is denoted by ${A^\theta }$.
Thus, $\overline {\left( {{A^T}} \right)} $ = ${\left( {\overline A } \right)^T}$ = ${A^\theta }$
Example:
If $A = \left[ {\begin{array}{c} {1 + 2i}&{2 - 3i}&{3 + 4i} \\ {4 - 5i}&{5 + 6i}&{6 - 7i} \\ 8&{7 + 8i}&7 \end{array}} \right]$,
Then ${A^\theta } = \left[ {\begin{array}{c} {1 - 2i}&{4 + 5i}&8 \\ {2 + 3i}&{5 - 6i}&{7 - 8i} \\ {3 - 4i}&{6 + 7i}&7 \end{array}} \right]$
