## beketso Group Title linear combinations on vectors.help one year ago one year ago

1. beketso Group Title

write the matrix$E= \left[\begin{matrix}3 & 1 \\ 1 & -1\end{matrix}\right]$

2. beketso Group Title

as a linear combination of the matrices $A=\left[\begin{matrix}1 & 1 \\ 1& 0\end{matrix}\right]$

3. beketso Group Title

$B=\left[\begin{matrix}0 & 0 \\ 1 & 1\end{matrix}\right]$

4. beketso Group Title

$C=\left[\begin{matrix}0 & 2 \\ 0 & -1\end{matrix}\right]$

5. richyw Group Title

are there just three?

6. beketso Group Title

basically ,you write the matrix E as a linear combination of the matrices A,B, and C

7. beketso Group Title

@richyw ,yes there only three

8. richyw Group Title

ok so the concepts you need here are matrix addition and scalar multiplication.

9. richyw Group Title

you can solve this one in different ways but it's simple enough to do by inspection

10. richyw Group Title

so looking at E, the first thing I notice is that $$e_{11}=3$$ now looking at matrices A, B, and C. The only one that has anything but 0 in that position is matrix A, so we know already that it must be 3 times matrix A. which gives. $\left[\begin{matrix}3 & 3 \\ 3 & 0\end{matrix}\right]$ right?

11. beketso Group Title

ok i got that part

12. richyw Group Title

alright so now look at matrix E and notice that position $$e_{12}$$ is 2, so we need to subtract a certain amount from matrix A to produce a 2 there.

13. richyw Group Title

well, matrix B has a zero in that position, so it is no good, so subtract one times matrix C from matrix A and we get $3A-C=\left[\begin{matrix}3&1\\3&1\end{matrix}\right]$

14. richyw Group Title

sorry I should have said "subtract one times matrix C from three times matrix A". Now I hope you can see what multiple of matrix B you bust subtract to get E

15. richyw Group Title

are you following?

16. beketso Group Title

yeah

17. richyw Group Title

alright cool well then you have the answer!

18. beketso Group Title

so it is 3A-2B-C=E?

19. richyw Group Title

yes!

20. beketso Group Title

thanx a lot!!!!!!!