Tamil Nadu Board of Secondary EducationHSC Arts Class 12

# If A = [53-1-2], show that A2 – 3A – 7I2 = O2. Hence find A–1 - Mathematics

Sum

If A = [(5, 3),(-1, -2)], show that A2 – 3A – 7I2 = O2. Hence find A–1

#### Solution

A = [(5, 3),(-1, -2)]

A2 = A × A

= [(5, 3),(-1, -2)][(5, 3),(-1, -2)]

= [(25 - 3, 15 - 6),(-5 + 2, -3 + 4)]

= [(22, 9),(-3, 1)]

– 3A = -3[(5, 3),(-1, -2)]

= [(-15, -9),(3, 6)]

– 7I2 = -7[(1, 0),(0, 1)]

= [(-7, 0),(0, -7)]

A2 – 3A – 7I2

= [(22, 9),(-3, 1)] + [(15, -9),(3, 6)] + [(-7, 0),(0, -7)]

= [(0, 0),(0, 0)]

∴ A2 – 3A – 7I2 = O2

Post multiply this equation by A–1

A2A– 3A A– 7IA1 = 0

A – 3I – 7A1 = 0

A – 3I = 7 A

A1 = 1/7 ("A" - 3"I")

= 1/7 [((5, 3),(-1, -2)), -3((1, 0),(0, 1))]

= 1/7 [((5, 3),(-1, -2)) + ((-3, 0),(0, -3))]

A1 = 1/7[(2, 3),(-1, -5)]

Concept: Inverse of a Non-singular Square Matrix
Is there an error in this question or solution?
Chapter 1: Applications of Matrices and Determinants - Exercise 1.1 [Page 15]

#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 12th Mathematics Volume 1 and 2 Answers Guide
Chapter 1 Applications of Matrices and Determinants
Exercise 1.1 | Q 4 | Page 15
Share