Maths > Matrices and Determinants > 3.0 Special Matrices
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
3.9 (i) Involuntary 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:
A matrix such that ${A^2} = I$ is called an involuntary matrix.
Question 7.
Show that the matrix $A = \left[ {\begin{array}{c} { - 5}&{ - 8}&0 \\ 3&5&0 \\ 1&2&{ - 1} \end{array}} \right]$ is involuntary.
Solution:
${A^2} = A.A = \left[ {\begin{array}{c} { - 5}&{ - 8}&0 \\ 3&5&0 \\ 1&2&{ - 1} \end{array}} \right] \times \left[ {\begin{array}{c} { - 5}&{ - 8}&0 \\ 3&5&0 \\ 1&2&{ - 1} \end{array}} \right]$
$ = \left[ {\begin{array}{c} {25 - 24 + 0}&{40 - 40 + 0}&{0 + 0 + 0} \\ { - 15 + 15 + 0}&{ - 24 + 25 + 0}&{0 + 0 + 0} \\ { - 5 + 6 - 1}&{ - 8 + 10 - 2}&{0 + 0 + 1} \end{array}} \right]$
$ = \left[ {\begin{array}{c} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right]$
$ = I$
Hence the given matrix $A$ is involuntary.
