Understanding Matrix Multiplication: Solving Homework Problems (5) and (24)

[Michael_Light]

Homework Statement

Homework Equations

The Attempt at a Solution

I need help with (5) and (24). For (5), i can find MU and MV but have difficulty in finding MⁿU and MⁿV. For (24), i can solve (i) but don't know how to find (A+I)²¹B.

The answer for (5): MⁿU=6ⁿu; MⁿV=9ⁿV...

As for (24) (A+I)²¹B =
row1: (-3 1 5)
row2: (6 -2 -10)
row3: (3 -1 -5)

Thanks in advance...

 

[vela]

For #5, try calculating MU and MV. Can you express the results in terms of U and V?

For #24, what did you get for (A+I)B? How is it related to B?

 

[Michael_Light]

For #5, try calculating MU and MV. Can you express the results in terms of U and V?

For #24, what did you get for (A+I)B? How is it related to B?

For (5), i get MU=
row1: (6)
row2: (6) and

MV=
(-9)
(18) and i stuck there...For (24), i get (A+I)B=
row1: (-3 1 5)
row2: (6 -2 -10)
row3: (3 -1 -5)
which is same as the answer of (A+I)²¹B... but i totally don't understand why.. can you help me?

P/S: i managed to find solve (5) already... can you focus on (24)? thanks...

 

[vela]

For (5), i get MU=
row1: (6)
row2: (6) and

MV=
(-9)
(18) and i stuck there...

OK, so you have

$$MU = \begin{pmatrix} 6 \\ 6 \end{pmatrix} = 6\begin{pmatrix} 1 \\ 1 \end{pmatrix} = 6 U$$

Now if you multiply by M again, you'll get

$$M^2 U = M(MU) = M(6U) = 6 (MU) = \dots$$

Can you see how this will work out?

For (24), i get (A+I)B=
row1: (-3 1 5)
row2: (6 -2 -10)
row3: (3 -1 -5)
which is same as the answer of (A+I)²¹B... but i totally don't understand why.. can you help me?

How is that matrix, (A+I)B, related to B?

 

[Michael_Light]

OK, so you have

$$MU = \begin{pmatrix} 6 \\ 6 \end{pmatrix} = 6\begin{pmatrix} 1 \\ 1 \end{pmatrix} = 6 U$$

Now if you multiply by M again, you'll get

$$M^2 U = M(MU) = M(6U) = 6 (MU) = \dots$$

Can you see how this will work out?

How is that matrix, (A+I)B, related to B?

(A+I)B = B.. okay i understand now... thanks for your help!