The sum of two idempotent matrices A and B is idempotent and if AB + B...
Given A2 = A and B2 = B
and (A + B) = ( A + B)2 = A2+ B2 + AB + BA
= A + B + AB + BA
⇒ A + B = A + B + AB + BA
⇒ AB + BA = 0 (0: Null matrix)
⇒ C = 0 ⇒ det C =0
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The sum of two idempotent matrices A and B is idempotent and if AB + B...
Idempotent Matrices
- An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix.
- Mathematically, for a matrix A to be idempotent, it must satisfy the condition A^2 = A.
Sum of Idempotent Matrices
- Let A and B be two idempotent matrices.
- We need to show that the sum of A and B, denoted as C = A + B, is also idempotent.
- To prove this, we can calculate the square of C: C^2 = (A + B)^2.
Calculating (A + B)^2
- Expanding (A + B)^2 using the distributive property, we get:
C^2 = (A + B)(A + B) = A(A + B) + B(A + B).
- Using matrix multiplication rules, we can further simplify this expression:
C^2 = A^2 + AB + BA + B^2.
AB = BA
- Given that AB = BA, we can substitute this into the expression for C^2:
C^2 = A^2 + AB + BA + B^2 = A + AB + BA + B = C + C.
Proving C is Idempotent
- From the equation C^2 = C + C, we can rearrange to obtain:
C^2 - C - C = 0.
- Factoring out C, we have:
C(C - I) - C = 0,
C(C - I - 1) = 0,
C(C - 2I) = 0.
Determinant of C
- The determinant of a matrix represents the scaling factor of the matrix.
- If the determinant is zero, it means the matrix is singular and does not have an inverse.
Conclusion: Determinant of C
- From the equation C(C - 2I) = 0, we can conclude that either C or (C - 2I) must have a determinant of zero.
- Therefore, the determinant of C is zero, as one of the factors in the product is zero.