LaraCroft said:Homework Statement
So, if the problem asks...
Find the elementary matrices such that the respective matrix equation holds...what do I do?
[_ _ _] [4 4 -1] [4 4 -1 ]
[_ _ _] [4 -1 -5] = [4 -1 -5]
[_ _ _] [-4 1 4] [-12 3 12]
Does this mean to find a matrix that when multiplied with the other gives the matrix on the other side of the equal sign?
Homework Equations
The Attempt at a Solution
Elementary matrices are square matrices with a single non-zero entry in each row and column, and all other entries being zero. They are used in linear algebra to perform elementary row operations, such as swapping rows, multiplying rows by a scalar, and adding a multiple of one row to another.
Writing A as a product of elementary matrices means expressing matrix A as a combination of elementary matrices that, when multiplied together, result in the original matrix A. This is useful for performing matrix operations and simplifying calculations.
Writing A as a product of elementary matrices allows us to easily perform operations on the matrix, such as finding its inverse or calculating its determinant. It also helps to simplify complex matrices and make them easier to work with.
To write A as a product of elementary matrices, you can use Gaussian elimination or row reduction to convert A into an identity matrix. The elementary matrices used in this process can then be multiplied together to form a product that results in A.
No, not all matrices can be written as a product of elementary matrices. This is because not all matrices can be reduced to an identity matrix using elementary row operations. However, most matrices encountered in linear algebra can be written as a product of elementary matrices.