If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I.

Home » If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I.

If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I.

We realize that,

$A\text{ }.\text{ }I\text{ }=\text{ }I\text{ }.\text{ }A$

In this way,

$A\text{ }and\text{ }I$

are commutative.

Subsequently, we can grow

${{\left( I\text{ }+\text{ }A \right)}^{3}}$

like genuine numbers development.

In this way,

${{\left( I\text{ }+\text{ }A \right)}^{3}}~=\text{ }{{I}^{3~}}+\text{ }3{{I}^{2}}A\text{ }+\text{ }3I{{A}^{2}}~+\text{ }{{A}^{3}}$

$=\text{ }I\text{ }+\text{ }3IA\text{ }+\text{ }3{{A}^{2~}}+\text{ }A{{A}^{2}}$

$(As\text{ }{{I}^{n~}}=\text{ }I,\text{ }n\text{ }\in \text{ }N)$

$=\text{ }I\text{ }+\text{ }3A\text{ }+\text{ }3A\text{ }+\text{ }AA$

$=\text{ }I\text{ }+\text{ }3A\text{ }+\text{ }3A\text{ }+\text{ }{{A}^{2}}$

$=\text{ }I\text{ }+\text{ }3A\text{ }+\text{ }3A\text{ }+\text{ }A\text{ }=\text{ }I\text{ }+\text{ }7A$

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