Question #254030

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

] .


1
Expert's answer
2021-11-07T16:41:50-0500

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

And express as follows:


M=[4779]=a[1111]+b[1234]+c[1145]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=[4779]=2[1111]+3[1234]1[1145]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.



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