Linear Algebra Answers

Compute all the minors and cofactors of

1 2 3

2 0 1

2 3 4

(6.1) Find the values of a, b and c such the matrix below is skew symmetric.

0 0 d

0 2a − 3b + c 3a − 5b + 5c

2 0 5a − 8b + 6c

(6.2) Give an example of a skew symmetric matrix.

(6.3) Prove that A (4) 2 is symmetric whenever A is skewsymmetric.

(6.4) Determine an expression for det(A) in terms of det(A (T ) if A is a square skewsymmetric.

(6.5) Assume that A has an odd number of rows and also an odd number of columns. In this particular case, show that det(·) is an odd function

Consider the matrices A = −2 7 1

3 4 1

8 1 5 ,

B = 8 1 5

3 4 1

−2 7 1 ,

C = −2 7 1

3 4 1

2 −7 3 .

Find elementary matrices E1, E2 and E3 such that

(5.1) E1A = B,

(5.2) E1B = A,

(5.3) E2A = C,

(5.4) E3C = A.

Determine whether or not the following matrices are in row echelon form or not? (4.1)

(1 2 −2)

( 0 1 2)

( 0 0 1)

(4.2) (1 2 −2)

(0 1 2 )

( 0 0 1 ) 

Determine whether or not the following matrices are in row echelon form or not? ( 12-2 012 001) (12-2 012 001)

Compute all the minors is and cofactors of (123 201 234)

Consider the matrices

A = −2 7 1

3 4 1

8 1 5,

B =8 1 5

3 4 1

−2 7 1,

C = −2 7 1

3 4 1

2 −7 3

Find elementary matrices E1, E2 and E3 such that

(5.1) E1A = B,

(5.2) E1B = A,

(5.3) E2A = C,

(5.4) E3C = A.

Change the following equations in to augmented matrix

x-y+2z=1

3x-y+5z=-2

4x+2y+(x²-8)z=(x+2)

And determine values of x where:-

There is no solution

And where there is exactly one solution

And where there infinitely many solutions

Suppose U1,U2,..,Um are finite-dimensional subspace of V.

Prove that :

U1+U2+...+Um is finite dementional and

dim(U1+U2+...+Um)≤ dimU1 + dimU2 + ..... + dimUm

Assume that T is an n×n matrix with a row of zeros .Prove that T has no inverse