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Matrices

Question
CBSEENMA12035689

If A is a square matrix such that straight A squared equals straight A comma then write the value of 7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed comma where I is an identity matrix.

Solution

Given that straight A squared equals straight A
We need to find the value of 7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed comma Where I is the identity matrix.
Thus, 
7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed space equals space 7 straight A minus open parentheses straight I cubed plus 3 straight I squared straight A plus 3 IA squared plus straight A cubed close parentheses
space space rightwards double arrow space 7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed space equals space 7 straight A minus open parentheses straight I cubed plus 3 straight A plus 3 straight A squared plus straight A squared cross times straight A close parentheses space space open square brackets straight I cubed space equals space straight I comma space straight I squared straight A equals straight A comma space IA squared equals straight A squared close square brackets
space space rightwards double arrow space 7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed equals 7 straight A minus open parentheses straight I plus 3 straight A plus 3 straight A plus straight A close parentheses space space space open square brackets because straight A squared equals straight A close square brackets
space space rightwards double arrow 7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed equals 7 straight A minus straight I minus 3 straight A minus 3 straight A minus straight A
space space space rightwards double arrow 7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed equals 7 straight A minus straight I minus 7 straight A
space space space space rightwards double arrow 7 straight A minus left parenthesis straight I plus straight A right parenthesis cubed equals negative straight I

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