Question

If P is a square matrix with P2 = P and if I is the unit matrix of the same order as of P then {P+1)4 =​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

P is a square matrix such that

$\rm :\longmapsto\: {P}^{2} = P$

and I is the identity matrix of same order as that of P.

Consider,

$\rm :\longmapsto\: {(P + I)}^{2}$

$\rm \:  =  \:  \: (P + I)(P + I)$

$\rm \:  =  \:  \: {P}^{2} + PI + IP + {I}^{2}$

We know,

$\boxed{ \bf{ \: AI = IA = A}}$

and

$\boxed{ \bf{ \: {I}^{2} = I}}$

$\rm \:  =  \:  \: P + P + P + I$

$\rm \:  =  \:  \: 3P + I$

$\bf\implies \: {(P + I)}^{2} =  3P + I$

Consider,

$\rm :\longmapsto\: {(P + I)}^{3}$

$\rm \:  =  \:  \: {(P + I)}^{2} (P + I)$

$\rm \:  =  \:  \: (3P + I)(P + I)$

$\red{\bigg \{ \because \: {(P + I)}^{2} = 3P + I \bigg \}}$

$\rm \:  =  \:  \: {3P}^{2} + 3PI + IP + {I}^{2}$

We know,

$\boxed{ \bf{ \: AI = IA = A}}$

and

$\boxed{ \bf{ \: {I}^{2} = I}}$

$\rm \:  =  \:  \: 3P + 3P + P + I$

$\rm \:  =  \:  \: 7P + I$

$\bf\implies \: {(P + I)}^{3} = 7P + I$

Now, Consider

$\rm :\longmapsto\: {(P + I)}^{4}$

$\rm \:  =  \:  \: {(P + I)}^{3}(P + I)$

$\rm \:  =  \:  \: (7P + I)(P + I)$

$\rm \:  =  \:  \: {7P}^{2} + 7PI + IP + {I}^{2}$

We know,

$\boxed{ \bf{ \: AI = IA = A}}$

and

$\boxed{ \bf{ \: {I}^{2} = I}}$

So, we get

$\rm \:  =  \:  \: 7P + 7P + P + I$

$\rm \:  =  \:  \: I5P + I$

$\bf\implies \: {(P + I)}^{4} = 15P + I$

Additional Information :-

1. Matrix multiplication of two matrices A and B is defined if number of columns of pre multiplier is equals to number of rows of post multiplier.

2. Matrix multiplication may or may not be Commutative.

3. Matrix multiplication is Associative. i.e. A(BC) (AB)C

4. Matrix multiplication is Distributive, i.e. A(B + C) = AB + AC