Let Ax 0 be a homogeneous system of n linear equations in n

Let Ax = 0 be a homogeneous system of n linear equations in n unknowns, and let Q be an invertible n times n matrix. Prove that Ax = 0 has only the trivial solution if and only if (QA) x = 0 has only the trivial solution. Let A and B be row equivalent matrices. Show that the linear systems Ax = 0 and Bx=0 have the same solution set.

Solution

Solution:

1)Solution

A homogeneous system of linear equations Ax = 0 has only the trivial solution if and only if the matrix A is invertible . Similarly, the homogeneous system (QA)x = 0 has just the trivial solution if and only if the matrix QA is invertible Therefore, we have to show that A is invertible if and only if QA is invertible.

If A is invertible and Q is invertible, then QA is invertible as the product of two invertible matrices .On the other hand, if QA is invertible, then both Q and Q are invertible , which completes the proof.

2)AX = Y and the homogeneous system can be written as AX = 0.

Before going to give the relation between matrices and solving systems of linear equations it we note that matrices have a good deal of algebra associated with them.

If particular it is possible to multiply a matrix by a scalar so that cA is the matrix obtained form A by multiplying all of its elements by c.

The sum A + B of two matrices of the same size is matrix whose (i, j)th entry is the sum of the corresponding elements of A and B and the product of an m × n matrix A with an m × p matrix B is the m × p matrix AB with elements (AB)ij = Xn k=1 AikBkj .