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The document provides a mathematical example demonstrating that the matrix A satisfies the equation A² - 4A + I = O, and subsequently finds the inverse of A. It includes exercises related to finding the adjoint of matrices, verifying properties of determinants, and solving systems of linear equations. Additionally, it discusses applications of determinants and matrices in solving linear equations.

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15 views2 pages

Teha

The document provides a mathematical example demonstrating that the matrix A satisfies the equation A² - 4A + I = O, and subsequently finds the inverse of A. It includes exercises related to finding the adjoint of matrices, verifying properties of determinants, and solving systems of linear equations. Additionally, it discusses applications of determinants and matrices in solving linear equations.

Uploaded by

dineshwar8c
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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92 MATHEMATICS

 2 3
Example 15 Show that the matrix A =   satisfies the equation A2 – 4A + I = O,
1 2
where I is 2 × 2 identity matrix and O is 2 × 2 zero matrix. Using this equation, find A–1.
 2 3   2 3   7 12 
Solution We have A 2 = A.A =    = 
1 2  1 2  4 7 
7 12   8 12  1 0   0 0 
Hence A 2 − 4A + I =  −  +  = =O
 4 7   4 8  0 1  0 0
Now A2 – 4A + I = O
Therefore A A – 4A = – I
or A A (A–1) – 4 A A–1 = – I A–1 (Post multiplying by A–1 because |A| ≠ 0)
or A (A A–1) – 4I = – A–1
or AI – 4I = – A–1
4 0  2 3  2 −3
or A–1 = 4I – A =   −  =  
0 4 1 2  −1 2 
 2 −3 
Hence A −1 =  
 −1 2 

EXERCISE 4.4
Find adjoint of each of the matrices in Exercises 1 and 2.
 1 −1 2
1 2  
1. 3 4  2.  2 3 5
   −2 0 1 
Verify A (adj A) = (adj A) A = | A | I in Exercises 3 and 4
1 −1 2 
2 3  
3.  −4 −6 4. 3 0 −2
  1 0 3 
Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
1 2 3
 2 −2  −1 5  
5. 4 3  6.   7. 0 2 4
   −3 2 0 0 5 

Reprint 2024-25
DETERMINANTS 93

1 0 0   2 1 3 1 −1 2 
3 3 0     
8.   9.  4 −1 0 10. 0 2 −3
5 2 −1  −7 2 1  3 −2 4 

1 0 0 
0 cos α sin α 
11. 
0 sin α − cos α 
3 7  6 8
12. Let A =   and B =   . Verify that (AB) = B A .
–1 –1 –1

 2 5   7 9 
 3 1
13. If A =   , show that A – 5A + 7I = O. Hence find A .
2 –1

 −1 2 
3 2 
14. For the matrix A =   , find the numbers a and b such that A2 + aA + bI = O.
1 1 

1 1 1 
 
15. For the matrix A = 1 2 −3
 2 −1 3 
Show that A3– 6A2 + 5A + 11 I = O. Hence, find A–1.
 2 −1 1 
 
16. If A =  −1 2 −1
 1 −1 2 
Verify that A3 – 6A2 + 9A – 4I = O and hence find A–1
17. Let A be a nonsingular square matrix of order 3 × 3. Then | adj A | is equal to
(A) | A | (B) | A | 2 (C) | A | 3 (D) 3 | A |
–1
18. If A is an invertible matrix of order 2, then det (A ) is equal to
1
(A) det (A) (B) det (A) (C) 1 (D) 0

4.6 Applications of Determinants and Matrices


In this section, we shall discuss application of determinants and matrices for solving the
system of linear equations in two or three variables and for checking the consistency of
the system of linear equations.

Reprint 2024-25

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