![Example 26 - Show A = [2 3 1 2] satisfies A2 - 4A + I = O - Examples](https://d1avenlh0i1xmr.cloudfront.net/42db4e92-6d98-42b0-a43e-ad4ab6c61b82/slide65.jpg)
Example 26 - Chapter 4 Class 12 Determinants
Last updated at May 29, 2018 by Teachoo
Transcript
Example 26 Show that the matrix A = 2312 satisfies the equation A2 – 4A + I = O, where I is 2 × 2 identity matrix and O is 2 × 2 zero matrix. Using this equation, find A–1. First calculating A2 A2 = A. A = 2312 2312 = 2 2+3(1)2 3+3(2)1 2+2(1)1 2+2(2) = 4+36+62+22+4 = 71247 Now, solving A2 – 4A + I Putting values = 71247 – 4 2312 + 1001 = 71247 – 4(2)4(3)4(1)4(2) + 1001 = 71247 – 81248 + 1001 = 7−8+112−12+04−4+07−8+1 = 8−812−124−48−8 = 0000 = O Thus, A2 – 4A + I = O Hence proved Now, Finding A-1 using equation A2 – 4A + I = O Post multiplying by A-1 both side (A2 – 4A +I) A-1 = OA-1 A2 . A-1 – AA-1 + I. A-1 = O A . (AA-1) – 4AA-1 + IA-1 = O A. I – 4I + IA-1 = O A – 4I + A-1 = O A-1 = O + A + 4I A-1 = A + 4I Putting values A-1 = – 2312 + 4 1001 = −2−3−1−2 + 4004 = −2+4−3+0−1+0−2+4 = 2−3−12 Thus, A-1 = 𝟐−𝟑−𝟏𝟐
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Chapter 4 Class 12 Determinants
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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.