1. Consider the multiplication of 2 x 2 (for simplicity) square matrix of determinant zero with an unknown 2 x 1 matrix yielding a non-zero matrix. Is one of the entries of the unknown matrix necessarily zero?
2. If so, will the result still hold
(a) if the unknown matrix has the same order as the square matrix, and
(b) for higher orders?
1. You consider a non-homogenous system AX=Y, where A is a 2x2 matrix, Y is a non-zero column vector and X is a column vector-solution. Is that?
In such case, there exist no solution OR infinity of solutions, because the determinant of A is zero. If we have the case of infinity solutions, there are between them such with 0 coordinate (entry). But we can not say that one of the entries IS zero.
Example 1: A has the rows 2, 1 and -4, -2. Y has coordinates 9, -18. The system has infinity solutions. One suitable vector X has coordinates 4, 1. There is no 0. Another solution has coord. 0, 9.
Example 2: A is that of Ex1, Y is 9, 9. The system has no solution.
2. Do you mind again AX=Y with a n x n matrix A and n x 1 column vectors X, Y?
Please, re-formulate.
A system of linear equation Ax = b is consistent if and only if b in column space of A.
Please see the attachment.
Dear Viera,
Thank you very much. If you are pointing out the implications of the first case, then you are on it. I figured out that my problem was the case of infinite solutions. And, for the second case, I was asking whether these conditions will still hold if X has the same order as A (n x n); the second part of this case is clear by the way.
Dear Prof Wiwat and John,
Thank you for the attachment and the example.
Dear Sanjeet,
it is the same for n-dimensional case. In general, it is not sure to have a solution with a zero coordinate.
However, if you add further conditions to the matrix AND to the right-side vector, there could appear necessarily only vector-solutions with a zero coordinate.
Geometrically: A zero coordinate signifies that the considered vector is orthogonal to the corresponding axis.
If a transformation Rn to Rn is defined by a matrix A, it is not a muss to obtain a desired vector Y from a vector X orthogonal to an axis.
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