3.7 (g) Idempotent 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 idempotent provided it satisfies the relation ${A^2} = A$.

Prove that the matrix $A = \left[ {\begin{array}{c} 2&{ - 2}&{ - 4} \\ { - 1}&3&4 \\ 1&{ - 2}&{ - 3} \end{array}} \right]$ is idempotent.

${A^2} = A.A = \left[ {\begin{array}{c} 2&{ - 2}&{ - 4} \\ { - 1}&3&4 \\ 1&{ - 2}&{ - 3} \end{array}} \right] \times \left[ {\begin{array}{c} 2&{ - 2}&{ - 4} \\ { - 1}&3&4 \\ 1&{ - 2}&{ - 3} \end{array}} \right]$

$ = \left[ {\begin{array}{c} {2.2 + \left( { - 2} \right).\left( { - 1} \right) + \left( { - 4} \right).1}&{2\left( { - 2} \right) + \left( { - 2} \right).3 + \left( { - 4} \right).\left( { - 2} \right)}&{2\left( { - 4} \right) + \left( { - 2} \right).4 + \left( { - 4} \right).\left( { - 3} \right)} \\ {\left( { - 1} \right).2 + 3.\left( { - 1} \right) + 4.1}&{\left( { - 1} \right).\left( { - 2} \right) + 3.3 + 4.\left( { - 2} \right)}&{\left( { - 1} \right).\left( { - 4} \right) + 3.4 + 4.\left( { - 3} \right)} \\ {1.2 + \left( { - 2} \right).\left( { - 1} \right) + \left( { - 3} \right).1}&{1.\left( { - 2} \right) + \left( { - 2} \right).3 + \left( { - 3} \right).\left( { - 2} \right)}&{1.\left( { - 4} \right) + \left( { - 2} \right).4 + \left( { - 3} \right).\left( { - 3} \right)} \end{array}} \right]$

$ = \left[ {\begin{array}{c} 2&{ - 2}&{ - 4} \\ { - 1}&3&4 \\ 1&{ - 2}&{ - 3} \end{array}} \right] = A$

Hence the matrix $A$ is idempotent.