Question

Let A =( 1 0 0
2 1 0
3 2 1
.if u1 and u2 are column matrices such that
Au1= 1
0
0
and Au2 = 0
1
0,then u1+u2 is equal to
opt
1.-1 2.-1
1 -1
-1 0
3. 1
-1
-1
4. -1
1
0

Answer 1

Given: A = [(1  0  0), (2  1  0), (3  2  1)], Au1 = [(1), (0), (0)] and Au2 = [(0), (1), (0)]

To find: u1 + u2 is equal to?

Solution:

• Now we have given a matrix: A = [(1  0  0), (2  1  0), (3  2  1)] and two column matrix as Au1 = [(1), (0), (0)] and Au2 = [(0), (1), (0)]

• Now since, we have given both A u1 and A u2, so adding them, we get:

       Au1 + Au2 = [(1), (0), (0)] + [(0), (1), (0)]

                         = [ 1 , 1 , 0 ]

• Since A is non singular matrix, so multiplying both sides by A^-1, we get:

          A^-1 x A(u1 + u2) = A^-1  [ 1 , 1 , 0 ]

          u1 + u2 = [(1  0  0), (2  1  0), (3  2  1)]^-1 x [ 1 , 1 , 0 ]

• Now, determinant of A is 1.

• After solving, adj of A comes out to be:

         [(1  0  0), (-2  1  0), (1  -2  1)]

         A^-1 = adj(A) / det(A)

                = [(1  0  0), (-2  1  0), (1  -2  1)]

•   So now putting it u1 + u2, we get:

        u1 + u2 = [(1  0  0), (2  1  0), (3  2  1)]^-1 x [ 1 , 1 , 0 ]

        u1 + u2 =  [(1  0  0), (-2  1  0), (1  -2  1)] x [ 1 , 1 , 0 ]

        u1 + u2 =  [(1 + 0 + 0), (-2 + 1 + 0), (1 - 2 + 0)]

        u1 + u2 = [1 , -1 , -1]

Answer:

      So the value of u1 + u2 is equal to [1 , -1 , -1] i.e third option.