3.10 (j) Nilpotent Matrix:

1.1 Type of Matrices
1.2 Trace of a Matrix:
1.3 Transpose of a Matrix
1.4 Conjugate of a Matrix

2.1 (a) Addition and subtraction of Matrices:
2.2 (b) Multiplication of 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:

4.1 Minors & Cofactors
4.2 Properties of Determinant

5.1 Elementary row Transformations

6.1 Properties of Inverse of a Matrix
6.2 Elementary row Transformations

8.1 Echelon form of a Matrix
8.2 Rank of a Matrix

A square matrix $A$ is called a nilpotent matrix if there exists a positive integer $k$ such that ${A^k} = 0$.If $k$ is the least positive integer such that ${A^k} = 0$ , then $k$ is called the index of the nilpotent matrix $A$

Show that $\left[ {\begin{array}{c} 1&1&3 \\ 5&2&6 \\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]$ is nilpotent matrix of order 3.

Let $A = \left[ {\begin{array}{c} 1&1&3 \\ 5&2&6 \\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]$

${A^2} = A.A = \left[ {\begin{array}{c} 1&1&3 \\ 5&2&6 \\ { - 2}&{ - 1}&{ - 3} \end{array}} \right] \times \left[ {\begin{array}{c} 1&1&3 \\ 5&2&6 \\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]$

$ = \left[ {\begin{array}{c} {1 + 5 - 6}&{1 + 2 - 3}&{3 + 6 - 9} \\ {5 + 10 - 12}&{5 + 4 - 6}&{15 + 12 - 18} \\ { - 2 - 5 + 6}&{ - 2 - 2 + 3}&{ - 6 - 6 + 9} \end{array}} \right]$

$ = \left[ {\begin{array}{c} 0&0&0 \\ 3&3&9 \\ { - 1}&{ - 1}&{ - 3} \end{array}} \right]$

$\therefore {A^3} = {A^2}.A = \left[ {\begin{array}{c} 0&0&0 \\ 3&3&9 \\ { - 1}&{ - 1}&{ - 3} \end{array}} \right] \times \left[ {\begin{array}{c} 1&1&3 \\ 5&2&6 \\ { - 2}&{ - 1}&{ - 3} \end{array}} \right]$

$ = \left[ {\begin{array}{c} {0 + 0 + 0}&{0 + 0 + 0}&{0 + 0 + 0} \\ {3 + 15 - 18}&{3 + 6 - 9}&{9 + 18 - 37} \\ { - 1 - 5 + 6}&{ - 1 - 2 + 3}&{ - 3 - 6 + 9} \end{array}} \right]$

$ = \left[ {\begin{array}{c} 0&0&0 \\ 0&0&0 \\ 0&0&0 \end{array}} \right]$

$ = 0$

$\therefore {A^3} = 0$ i.e., ${A^k} = 0$

Here $k = 3$.

Hence $A$ is nilpotent of order 3.