Matrices and Determinant

Jason Leav

Matrices

1. Compute AB ,ifA= [

] [

]

2. [

] [

]

a. Calculate AB and BA .Hence evaluate

b. Show that for any number k,( ) where I is the unit matrix.

3. If [

] choose and so that ( )

4. If [ ] [ ] and C [

]verify that ( ) where T denote

by transpose

5. Write the following transformation in matrix form: √

√

.

Hence, find the transformation in matrix form which expresses in terms of

6. If [

] and I is a unit matrix, show that I+A=( ) [ s si si s

].

7. If A=[ s si si s

] ,then show that [ s si si s

]

8. Show that [ s si

si s ]=[

] [

]

9. Verify

[

] is and orthogonal.

10. Show that [ s si

s s si s

] is and orthogonal.

11. Show that A [

] is orthogonal matrix.

12. Prove that ( ) if

13. If A, B and C are matrices such that the products of AC and BC obey the commutative law

prove that ( ) ( ).

14. If a matrix A satisfies a relation prove that exists and that ,I

being an identity matrix.

15. If A and B are non-singular matrices of the same order then ( ) Hence,prove

that ( ) ( ) for any positive integer m.

16. Prove that the inverse of a matrix is unique.

17. Find the adjoint and inverse of the following matrices : (1-6)

1.[

] 2. [

] 3. [

]

Matrices and Determinant

Jason Leav

4.[

] 5. [

] 6. [

]

18. Proof Diagonalisation of a matrix.

19. If the columns of a matrix A mxn type and the rows of a matrix B n-

type it. p. Show that

20. Show that if A is an invertible matrix, O (A is also good for every number a non-zero

( )

.

21. Show that if two invertible matrices A and B commute, then also commute.

Matrices and Determinant

Jason Leav

Determinant

1. Minors and Cofactors

A. A=|

|,B=|

| |

| |

| |

|

B. Expand the determinant

|

|, |

|, |

| |

|and|

|

C. Expand the following determinants by two methods:

1. Along the-third row

2. Along the-third column

|

|, |

|, |

| |

|and|

g g g

|

2. Properties of determinants

A. Proof that |

| |

|

B. Proof that |

| |

|(also changing the column).

C. Proof that , where |

|and |

|

D. Proof that |

|=|

|

E. Using the properties of determinants, Determine:

|

|,|

| |

|,

|

|,|

|

F. If is the one of the imaginary cube roots of unity ,find the value of the determinant

A=|

|

G. Without expanding the determinant, prove that ( )is a factor of |

|

H. Evaluate |

|

Matrices and Determinant

Jason Leav

I. Without expanding the determinant, prove that |

| ( ),

|

| ,||

|| |

|,|

|

( )( )( )( ) |

|

J. Prove that|

|=|

| |

|+|

|

K. Without expanding the determinant, prove that |

|

( )( ) |

| |

|

L. Show that |

| |

|,|

|

3. Factory Theorem: Show that |

| ( )( )( ) |

|

( )( )( ), |

|=( )( )( )( )

4. Laplace method proof.

5. A Pr f the Cramer’s ru e

B. Solve the f wi g system f equati usi g Cramer’s ru e: