Question

It A is square matry that A²=A, then find
ther alue of inand If
(I+ A) 3 - 7A)
(m and n ccontants) ​

Answer 1

Basic Concept Used :-

The properties of Identity Matrix :-

$\red{ \: \rm :\longmapsto\:AI = IA = A}$

$\red{ \: \rm :\longmapsto\: I \: I= I}$

Appropriate Question :-

$\red{\sf \:If \: A \: is \: a \: square \: matrix \: such \: that \: {A}^{2} = A, \: then} \\ \red{ \sf \: find \: the \: value \: of \: {(I + A)}^{3} - 7A \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

Solution :-

Given that,

$\blue{\bf :\longmapsto\: {A}^{2} = A}$

Now,

Consider,

$\red{\rm :\longmapsto\: {(I + A)}^{3} - 7A}$

$\rm \: = \: \: (I + A)(I + A)(I + A) - 7A$

$\rm \: = \: \: (I + IA + AI + {A}^{2})(I + A) - 7A$

$\rm \: = \: \: (I + A + A + A)(I + A) - 7A$

$\rm \: = \: \: (I + 3A)(I + A) - 7A$

$\rm \: = \: \: I + IA + 3AI + 3 {A}^{2} - 7A$

$\rm \: = \: \: I + A + 3A + 3A - 7A$

$\rm \: = \: \: I + 7A - 7A$

$\rm \: = \: \: I$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf{ {(I + A)}^{3} - 7A = I}}}$

Additional Information :-

1. Matrix multiplication may or may not be Commutative.

2. Matrix multiplication is Associative.

3. Matrix multiplication is Distributive.

4. Matrix multiplication, AB is possible only when number of columns of Matrix A is equal to number of rows of matrix B, otherwise AB is not defined.

Answer 2

Given Condition :

$\sf A^2 = A$

Such a matrix is known as Idempotent matrix.

Expanding the above expression,

$\sf (I + A)^3 - 7A \\ \\ \longrightarrow \sf I^3 + 3A^2I + 3AI^2 + A^3 - 7A \\ \\ \longrightarrow \sf I + 3A^2 + 3A - 7A \\ \\ \longrightarrow \sf AA^{-1} + 3A + 3A - 7A \\ \\ \longrightarrow \sf A^2A^{-1} - A \\ \\ \longrightarrow \sf A - A \\ \\ \longrightarrow \sf 0$

Thus,

$\boxed{\boxed{\sf (I + A)^3 - 7A = 0 }}$