let-matrix-A-n-n-and-B-n-n-satisfying-A-2-A-amp-B-2-B-then-prove-A-B-A-AB-B-AB-here-rank-

Question Number 206609 by universe last updated on 20/Apr/24

let matrix A_(n×n) and B_(n×n) satisfying A^2 = A & B^2 = B then prove ρ(A−B) = ρ(A−AB)+ ρ(B−AB) here ρ = rank

$$\:\:\:\mathrm{let}\:\mathrm{matrix}\:\:\mathrm{A}_{\mathrm{n}×\mathrm{n}} \:\mathrm{and}\:\mathrm{B}_{\mathrm{n}×\mathrm{n}} \:\mathrm{satisfying} \\ $$$$\:\:\:\mathrm{A}^{\mathrm{2}} =\:\mathrm{A}\:\:\&\:\:\mathrm{B}^{\mathrm{2}} =\:\mathrm{B}\:\:\mathrm{then}\:\mathrm{prove} \\ $$$$\:\:\:\rho\left(\mathrm{A}−\mathrm{B}\right)\:=\:\rho\left(\mathrm{A}−\mathrm{AB}\right)+\:\:\rho\left(\mathrm{B}−\mathrm{AB}\right) \\ $$$$\:\:\:\mathrm{here}\:\rho\:=\:\mathrm{rank} \\ $$

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