Suppose a 2 × 2 matrix A has an eigenvector (1 2) , with corresponding eigenvalue −4. what is A(-2 -4) ?

2 Answers
May 4, 2017

# A ( (-2, -4) ) = ( (8,16) ) #

Explanation:

If #lamda# is an eigenvalue with corresponding eigenvector #ul v# then:

# A ul v = lamda ul v #

From which we get with #lamda=-4# and #ul v = ( (1,2) )# :

# A ( (1,2) ) = -4 ( (1,2) ) \ \ \ ..... (star)#

A properties of matrices is that #Amu B = mu AB#, and so

# A ( (-2, -4) ) = (-2)A ( (1, 2) ) #
# " " = (-2)(-4) ( (1,2) ) # using #(star)#
# " " = 8 ( (1,2) ) #
# " " = ( (8,16) ) #

May 4, 2017

By definition, the eigenvectors (#mathbf e_i#) of #A#, and its eigenvalues (#lambda_i#) are related as:

#A mathbf e_i = lambda_i mathbf e_i#

Suppose a 2 × 2 matrix A has an eigenvector (1 2) , with corresponding eigenvalue −4. what is A(-2 -4) ?

This means that:

#A((1),( 2)) = - 4 ((1),( 2))#

So with the question what is A(-2 -4) ?:

#- 2 ( A((1),( 2)) )= - 2 ( - 4 ((1),( 2)))#

#implies A((-2),( -4)) = ((8),( 16))#