If A is a square matrix such that A2=I, then (A-I)3+(A+I)3-7 A is equal to [NCERT Exemplar]

Question

If A is a square matrix such that A2=I, then (A-I)3+(A+I)3-7 A is equal to [NCERT Exemplar] (A) A (B) l-A (C) 1+A (D) 3 A

Tardigrade Admin · Accepted Answer

Given that, A2=1 Consider, (A-I)3+(A+I)3-7 A =A3-I3-3 A2 l+3 A I2+A3+I3+3 A2 I+3 A I2-7 A [∵(a+b)3=a3+b3+3 a2 b+3 a b2 text and (a-b)3=a3-b3-3 a2 b+3 a b2] =2 A3+6 A I2-7 A =2 A2 ⋅ A+6 A I-7 A (∵ A3=A2 ⋅ A. and .I2=I) =2 l ⋅ A+6 A-7 A(∵ A2=l) =2 A+6 A-7 A =8 A-7 A =A

Q. If $A$ is a square matrix such that $A^{2} = I$ , then $(A - I)^{3} + (A + I)^{3} - 7 A$ is equal to [NCERT Exemplar]

$A$

$l - A$

$1 + A$

$3 A$

Q. If A is a square matrix such that A2=I, then (A−I)3+(A+I)3−7A is equal to [NCERT Exemplar]

A

l−A

1+A

3A

Given that, A2=1 Consider, (A−I)3+(A+I)3−7A =A3−I3−3A2l+3AI2+A3+I3+3A2I+3AI2−7A [∵(a+b)3=a3+b3+3a2b+3ab2 and (a−b)3=a3−b3−3a2b+3ab2​] =2A3+6AI2−7A =2A2⋅A+6AI−7A(∵A3=A2⋅A and I2=I) =2l⋅A+6A−7A(∵A2=l) =2A+6A−7A =8A−7A =A

Q. If $A$ is a square matrix such that $A^{2} = I$ , then $(A - I)^{3} + (A + I)^{3} - 7 A$ is equal to [NCERT Exemplar]

$A$

$l - A$

$1 + A$

$3 A$