Search
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.
