Let A be a matrix such that \(A^2 = I\), where I is an identity matrix, then \((I + A)^4 - 8A\) is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(8I\)
Since I commutes with every matrix, \((I + A)^2\) can be expanded like a square of numbers. Using \(A^2 = I\):
\((I + A)^2 = I + 2A + A^2\)
\(= I + 2A + I\)
\(= 2(I + A)\)
Squaring again:
\((I + A)^4 = 4(I + A)^2\)
\(= 4 \times 2(I + A)\)
\(= 8I + 8A\)
Hence
\((I + A)^4 - 8A = 8I\)