Video Transcript
True or False: The number of rows
of the augmented matrix of a system of linear equations equals the number of rows of
the coefficient matrix of this system.
In this question, we are asked if
the number of rows of an augmented matrix of a system of linear equations must be
equal to the number of rows of the coefficient matrix of the same system of
equations. We can do this by recalling what we
mean by the augmented matrix of a system of linear equations and what we mean by the
coefficient matrix of a system of linear equations.
We want to compare the matrices for
any system of linear equations. So we need a general system of
linear equations as shown. Each of the unknowns 𝑥 sub 𝑖 are
variables of the system. In this system, we have 𝑛
unknowns. Similarly, each of the values of 𝑎
sub 𝑖𝑗 are constant coefficients of the variables, and each value of 𝑏 sub 𝑖 is
a constant. We have 𝑚 equations in this system
of linear equations.
We can then recall that the
augmented matrix of this system is found by constructing a matrix with elements in
each row and column corresponding to the coefficients of the corresponding variables
of the system, with an added final column for the constants of the linear
equation. The number of rows of this matrix
must be equal to the number of linear equations, which is 𝑚. Similarly, the number of columns
will be equal to the number of variables in the system plus one for the column of
constants, that is, 𝑛 plus one.
The coefficient matrix of this
system is defined to be a matrix with elements in each row and column corresponding
to the coefficients of the corresponding variables of the system. This gives us the following
coefficient matrix of this system of equations. We can note that the augmented
matrix also defined its elements using the corresponding coefficients of the
variables in the system of equations. In particular, all of the entries
in the first 𝑚 rows and 𝑛 columns of the augmented matrix are equal to those in
the coefficient matrix.
So not only does the coefficient
matrix have 𝑚 rows and 𝑛 columns, meaning that the augmented matrix and
coefficient matrix have the same number of rows, we can also say that the entries in
the 𝑚-by-𝑛 submatrix of the augmented matrix by taking out the final column is
equal to the coefficient matrix.
Hence, the answer is that it’s true
that the number of rows in the augmented matrix of a system of linear equations must
be equal to the number of rows of the coefficient matrix of the same system of
equations.
