Q. 565.0( 2 Votes )

If A is a square

Answer :

We are given that,

A is a square matrix such that,


A2 = A


I is an identity matrix.


We need to find the value of 7A – (I + A)3.


Take,


7A – (I + A)3 = 7A – (I3 + A3 + 3I2A + 3IA2)


[, by algebraic identity, (x + y)3 = x3 + y3 + 3x2y + 3xy2]


7A – (I + A)3 = 7A – I3 – A3 – 3I2A – 3IA2


7A – (I + A)3 = 7A – I – A3 – 3I2A – 3IA2


7A – (I + A)3 = 7A – I – A.A2 – 3I2A – 3IA2


7A – (I + A)3 = 7A – I – A.A2 – 3A – 3A2


[, by property of identity matrix,


I2A = A & IA2 = A2]


7A – (I + A)3 = 7A – I – A.A – 3A – 3A


[, it is given that, A2 = A]


7A – (I + A)3 = 7A – I – A2 – 6A


[, A.A = A2]


7A – (I + A)3 = 7A – I – A – 6A


[, it is given that, A2 = A]


7A – (I + A)3 = 7A – I – 7A


7A – (I + A)3 = -I


Thus, the value of 7A – (I + A)3 is –I.


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