Linear Algebra Answers

Disprove that the commutative property holds on the matrix multiplication.

Prove that the commutative property holds on the matrix multiplication.

T :ℝ2 → ℝ2 as, 𝑇 𝑥, 𝑦 = (1, 𝑦) ;is it a linear transformation?

26. Reduce the Quadratic Form (Q.F.) 2x

2 + 5y

2 + 3z

2 + 4xy to canonical form by an

orthogonal transformation. Also find its nature, rank, index and signature of the Q.F.

Diagonalize the matrix





10 −2 −5

−2 2 3

−5 3 5





If 2 is one of the eigenvalue of





−2 2 −3

2 1 −6

−1 −2 0



 then find the other two eigenvalues.

2. Use Cayley-Hamilton theorem to find A6 − 5A5 + 8A4 − 2A3 − 9A2 + 31A − 36I,

when A=





1 0 3

2 1 −1

1 −1 1





10.) Consider the linear equation 2a + 3b = 4

Is (a; b) = ( 12 ; 1) a solution to the equation? Motivate your answer.

11.) Look up what is meant by a system of linear equations.

A known fact of solutions of systems of linear equations is that only one the following options can hold :

(a) No solution possible

(b) A unique solution can be found

(c) The system has infinite solutions.

Consider that two straight lines form a linear system.

Interpret what happens geometrically to the straight lines to get each case of the solution types given above.

12.) Look up the concept of a homogeneous linear system.

Only two solution types of the three mentioned solution types above are possible. Which one can never happen and why.

4.) True or False : 3Z = Z + Z + Z when Z is a matrix.

􏰀1 2􏰁 􏰀a􏰁

5.) Let X = 3 4 ; E = b

Find each of the following. If the operation cannot be done : state undefined operation.

a) XE

b) EX

c) XT X where XT stands for the transpose of X

10.) Consider the linear equation 2a + 3b = 4

Is (a; b) = ( 1/2 ; 1) a solution to the equation? Motivate your answer.

11.) Look up what is meant by a system of linear equations.

A known fact of solutions of systems of linear equations is that only one the following options can hold :

(a) No solution possible

(b) A unique solution can be found

(c) The system has infinite solutions.

Consider that two straight lines form a linear system.

Interpret what happens geometrically to the straight lines to get each case of the solution types given above.

12.) Look up the concept of a homogeneous linear system.

Only two solution types of the three mentioned solution types above are possible. Which one can never happen and why.