MAT1503 ASSIGNMENT 2 2025

MAT1503
ASSIGNMENT 2
2025

, QUESTION 1


1.1).


(A−1 + B−1 )−1 = A(A + B)−1 B


Multiple both side by (A−1 + B−1 ):


(A−1 + B−1 )−1 = A(A + B)−1 B
(A−1 + B−1 )−1 (A−1 + B −1 ) = (A(A + B)−1 B)(A−1 + B−1 )
I = A(A + B)−1 BA−1 + A(A + B)−1 BB−1 ∴ BB−1 = I
I = A(A + B)−1 BA−1 + A(A + B)−1 I
I = A(A + B)−1 (BA−1 + I)
I = A(A + B)−1 (I + BA−1 ) ∴ I = AA−1
I = A(A + B)−1 (AA−1 + BA−1 ) ∴ (AA−1 + BA−1 ) = (A + B)A−1
I = A(A + B)−1 (A + B)A−1
I = A[(A + B)−1 (A + B)]A−1
I = AIA−1
I = AA−1
I=I (Proven)




1.2).


A2 + 5A − I = 0
A2 + 5A = I
AA + 5A = I
A−1 (AA + 5A ) = A−1 I ∴ A−1 I = A−1
A−1 AA + A−1 (5A) = A−1
IA + 5A−1 A = A−1 ∴ A−1 A = I and IA = A

, A + 5I = A−1
A−1 = A + 5I




Clearly there is a mistake in the question, so the expression cannot be proved.


1.3).


1 −1
A=[ ]
−2 3


P(x) = 2x 2 − x + 1
P(A) = 2A2 − A + I

1 −1 2 1 −1 1 0
P(A) = 2 ([ ]) − [ ]+[ ]
−2 3 −2 3 0 1
1 −1 1 −1 1 −1 1 0
P(A) = 2 ([ ][ ]) − ([ ]−[ ])
−2 3 −2 3 −2 3 0 1
3 −4 0 −1
P(A) = 2 [ ]−[ ]
−8 11 −2 2
6 −8 0 −1
P(A) = [ ]−[ ]
−16 22 −2 2
6 −7
P(A) = [ ]
−14 20




1.4).


3 0 0
A = [0 −1 3 ]
0 −3 −1


P(x) = x 2 − 9
P(x) = (x − 3)(x + 3)