Answer to Question #254030 in Linear Algebra for usama

Question

Express M as a linear combination of the matrices A, B, C, where M = [

4 7

7 9

] , A = [

1 1

1 1

] , B

= [

1 2

3 4

] , C = [

1 1

4 5

] .

Answer 1

To show M as linear combination s of A B C.. we take three variable a,b,c

And express as follows:

$M=\begin{bmatrix}4\\7\\ 7\\ 9 \end{bmatrix}=a \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} +b \begin{bmatrix} 1\\ 2\\ 3\\ 4 \end{bmatrix} +c \begin{bmatrix} 1\\ 1\\ 4\\ 5 \end{bmatrix}$

Then we get 4 linear equations

4=1a+1b+1c...........1

7=1a+2b+1c...........2

7=1a+3b+4c............3

9=1a+4b+5c...........4

To solve:

Take eqn. 1 and 2

Subtracting 1 from 2 .

We get b=3.

Now putting value of b in eqn. 2 and 3 ..

Now subtracting 2 from 3.

We get value of c =-1

Now putting values of b and c in equation 4.

We get value of a= 2.

Thus,

By solving these 4 equation we get,

a=2

b=3

c=-1

Now ;

Putting back these values in matrix equation

$M=\begin{bmatrix}4\\7\\ 7\\ 9 \end{bmatrix}=2 \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} +3 \begin{bmatrix} 1\\ 2\\ 3\\ 4 \end{bmatrix} -1 \begin{bmatrix} 1\\ 1\\ 4\\ 5 \end{bmatrix}$

Thus we get expression for M as linear combination of A,B,C matrices.