Question

Please explain this concept of Linear algebra ( Matrices and Vector ) ?

Case 1 : When Matrix is multiplied with a Column Vector ie `Ax`
Case 2 : When a Scalar is multiplied to a Column Vector ie `lambdax`

If `Ax = lambdax` ;

Here the Matrix `A` and the scalar `lambda` are not equal (bcoz matrix and scalar are totally different things) then how their products with vector `x` is equal ??

!!! plz give some nice explanation !!!

Answer 1

See below.

Explanation:

The basic rule you need to understand is that when you multiply two matrices `A` and `B` you will obtain a third matrix `C` which is possibly different in size from both `A` and `B`.

The rule states that, if `A` is a `(n \times m)` matrix and `B` is a `(m \times p)` matrix, then `C` will be a `(n \times p)` matrix (note that the number of columns of `A` and the number of rows of `B` must be the same, in this case `m`, otherwise you can't multiply `A` and `B`).

Also, you can consider vectors as special matrices, having only one row (or column).

Let's say that in your case `A` is a `(n \times n)` matrix. It follows that `x` must be a column vector with `n` rows and one column. So, by the rule above, the product between `A` and `x` is of the form

`(n\times n)(n\times 1) = (n\times 1)`

And thus `Ax` has the same shape of `x` itself.

In the same way, `\lambda x` is just `x` multiplied by some constant, and thus its shape won't change.

So, being both vectors of the same shape `(n \times 1)`, it makes sense to ask if they are equal.

P.S. Note that it is necessary for `A` to be a square matrix. In fact, if `A` is a `(m \times n)` matrix, then `Ax` is a `(m\ times 1)` vector, and can't be a multiple of `x`.