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How to use augmented matrices to solve systems of linear equations through various elementary row operations. It covers interchanging equations, multiplying equations by nonzero constants, and adding constant multiples of one equation to another. The document also discusses the concept of matrices, including their size, identity matrices, and position of elements. The document concludes by forming an augmented matrix, solving it using elementary row operations, and finding the solution.
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Elementary Row Operations
To solve the linear system algebraically, these steps could be used.
x 5y z 11
3z 12
2x 4y 2z 8
All of the following operations yield a system which is equivalent to the original. (Equivalent
systems have the same solution.)
Interchange equations 2 and 3
x 5y z 11
2x 4y 2z 8
3z 12
Multiply equation 3 by
x 5y z 11
2x 4y 2z 8
z 1
Multiply equation 2 by
x 5y z 11
x 2y z 4
z 1
Add equation 1 to 2 and replace
x 5y z 11
3y 15
z 4
equation 2 with the result
Multiply equation 2 by
x 5y z 11
y 5
z 4
Multiply equation 2 by and add it 5
x z 14
y 5
z 4
to equation 1; replace equation 1 with
the result
Add equation 3 to equation 1; replace
x 18
y 5
z 4
equation 1 with the result
The solution is (18, −5, 4).
a) Two equations are interchanged.
b) An equation is multiplied by a nonzero constant.
c) A constant multiple of one equation is added to another equation.
Matrices
A matrix is a rectangular array of numbers written within brackets. The size of a matrix is always given
in terms of its number of rows and number of columns (in that order!). A 2 x 4 matrix has 2 rows
and 4 columns. Square matrices have the same number of rows and columns. A matrix with a single
column is called a column matrix, and a matrix with a single row is called a row matrix. A square
matrix with all elements on the main diagonal equal to 1 and all other elements equal to 0 is called an
identity matrix. The 3x3 identity matrix is.
The position of an element within a matrix is given by the row and column (in that order!) containing
the element. The element is in row 3 and column 4. 34
a
a) Two rows are interchanged i j
b) A row is multiplied by a nonzero constant i i
kR →R
c) A constant multiple of one row is added to another row j i i
kR + R →R
(NOTE : →means "replaces")
Forming an Augmented Matrix
An augmented matrix is associated with each linear system like
x 5y z 11
3z 12
2x 4y 2z 8
The matrix to the left of the bar is called the coefficient matrix.
To solve a system using an augmented matrix, we must use elementary row operations to change
the coefficient matrix to an identity matrix.
Form the augmented matrix
Interchange rows 2 and 3
2 3