Matrix multiplaction

i know how to multiply matrices but I don't know what to do for this question. What aij mean and what does a23 mean and how do I find it?

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Reply 1

IrrationalRoot

a_{ij}

is the element of

A

that is in row

i

and column

j

. So

a_{23}

is just the element in the second row and third column. To work out what it is... you say you know the definition of matrix multiplication, so use it. You'll get some equations, then use the ones that are relevant to what you want to find.

Reply 2

the bear

Original post by Kira Yagami

i know how to multiply matrices but I don't know what to do for this question. What aij mean and what does a23 mean and how do I find it?

so A is a 3 x 3 matrix.

they want you to work out the element which is on the second row and the third column

Reply 3

username1084917

Original post by IrrationalRoot

a_{ij}

is the element of

A

that is in row

i

and column

j

. So

a_{23}

ok, thanks. But how do I find A?

Reply 4

IrrationalRoot

Original post by Kira Yagami

ok, thanks. But how do I find A?

You don't need to find

A

; remember you only need

a_{23}

. So just assign letters to the elements of

A

(ideally the appropriate letters

a_{ij}

) and then put that into the equations you have to get equations relating these

a_{ij}

. You'll get nine equations (three for each of the original equations), but it should be clear that you don't need all of them to find

a_{23}

Reply 5

username1084917

Original post by IrrationalRoot

You don't need to find

A

; remember you only need

a_{23}

. So just assign letters to the elements of

A

(ideally the appropriate letters

a_{ij}

) and then put that into the equations you have to get equations relating these

a_{ij}

. You'll get nine equations (three for each of the original equations), but it should be clear that you don't need all of them to find

a_{23}

Okay, but would it not be easier to solve to find matrix A and then identify a23?

Thanks

Reply 6

IrrationalRoot

Original post by Kira Yagami

Okay, but would it not be easier to solve to find matrix A and then identify a23?

Thanks

Unless I'm missing something, finding

A

itself would involve finding each element individually, which is clearly more work than just finding one (or a few).

Reply 7

username1084917

Original post by IrrationalRoot

Unless I'm missing something, finding

A

itself would involve finding each element individually, which is clearly more work than just finding one (or a few).

I'm confused as to what equations I need to solve :/

Reply 8

IrrationalRoot

Original post by Kira Yagami

I'm confused as to what equations I need to solve :/

So just write the matrix

A

with a letter in each entry, then put it into your three equations. On the left hand side, you get this matrix of letters times a vector, so do that multiplication to get a vector (with letters in it). On the right hand sides, you just have a given vector. Then you can just equate the components of the left side vector with the right side vector.

Reply 9

IrrationalRoot

Alternatively to the method I suggested (and that is probably expected), you could be a bit clever and find a linear combination of the three vectors (that you're multiplying by

A

) that equals the vector

(0,0,1)^T

, but this probably isn't much quicker. If this doesn't make sense, feel free to ignore it; it's not important.

Reply 10

username1084917

Original post by IrrationalRoot

You don't need to find

A

; remember you only need

a_{23}

. So just assign letters to the elements of

A

(ideally the appropriate letters

a_{ij}

) and then put that into the equations you have to get equations relating these

a_{ij}

. You'll get nine equations (three for each of the original equations), but it should be clear that you don't need all of them to find

a_{23}

Would you mind giving me an example of one of the sets of equations I need solve, I'm still confused :/

Reply 11

Deranging

Let the

i, j

th entry of the matrix

A

a_{i j}

. So, if we let

A

be a

3 \times 3

matrix this would mean that

Unparseable latex formula:

A = \left( \begin{array}{c c c} a_{1 1} & a_{1 2} & a_{1 3} \\[br]a_{2 1} & a_{2 2} & a_{2 3} \\ a_{3 1} & a_{3 2} & a_{3 3} \end{array} \right).

This is an annoying amount of writing, so usually you abbreviate this

A = ( a_{i j} )

.
Similary, vectors also require an annoying amount of writing, so if you have a vector, you can similarly write it

\vec{v} = (v_i)

. This is a component notation and it is very useful, because it saves a lot of writing (and for many other reasons).
How would you write the components of the vector