Solving a system of linear equations is one of the cumbersome operations in mathematics. But it is very essential in mathematics. Here we consider the elimination method using matrices.
Give a system of the linear equations.
Next we multiply the equation by
and add it to
. Also multiply the equation
by
and add to
. Then we have
Now we apply elementary operations to the system of linear equations.
First of all,
The matrix composed of coefficients of the system of linear equations is called a coefficient matrix. The matrix composed of coefficient matrix and constant terms is called an augmented matrix and denoted by
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Generally, those 3 operations on a metrix is called fundamental row operation.
Answer The augmented matrix is given by
Now the Pivot element
. So, use the elementary operation
. Then
Thus, we have
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If a matrix is created by appling finitely many elementary operation on
, a matrix
is said to be row equivalent and denoted by
. If elmentary operation
or
is applied once to the identity matrix of the order
, then the matrix obtained is called a fundamental matrix. You might already have noticed that an elementary operation can be written using an elementary matrix. For example, the elementary operation from
to
is
and the corresponding elementary matrix can be obtained by applying the elementary operation
on
.
When you apply elementary operations on a matrix . we are able to obatin a matrix so that all entries below the diagonal are zero. We say this matrix as upper triangular matrix.
Proof It is up to you..
Answer
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In this example, the order of the row operation is not important.
Proof It is up to you.
It is important to know that the matrix row equivalent to is unique.
Rank of matrix
The number of steps of the row reduced echlon form is important for application. This number is called the rank of a matrix and denoted by
. For example, the rank of 2.2 is
.
Proof
If
, then the rank of
is
. Thus,
.
Conversely, if
, then the number of steps of the row reduced matrix of
is
. By the definition of the row reduced echlon matrix, the first nonzero entry is
. Then every diagonal element is
. Thus,
.
The rank of a matrix can be defined by the concept of a vector space.
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Proof
Let be a matrix of
. Let the row vectors of
be
. A row space is alinear combination of
Next, we use to find the row reduced echlon matrix. Suppose that
is a linear combination of
, then
and
are the same. This corresponds to zeor row vector in the row echlon matrix and removing the row vector
. Repeating this process, we can find the row vector
corresponding to the row reducing. The row vectors are linealy independent. Thus the dimension of the row vectors is the same as the rank of row reduced matrix.
The rank of a matrix plays an important role on solving the system of linear equations. Befor moving to the next section, try to solve the system of linear equations.
Answer
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1. Find the row reduced matrix which is row equivalent to
.
2. Find the rank of the following matrices.
3. Given
. Apply elmentary operations
.
4.
can be reduced to the identiry matrix by using the elementary row operation. Find the product of matrices
so that
.
5. Find the dimension of the subspace spanned by the following vectors.