Answer:
a) The answer is on the first uploaded image
b) The answer is on the second uploaded image
c) The answer is on the third uploaded image
d) The answer is on the fourth uploaded image
e) The answer is on the fifth uploaded image
Step-by-step explanation:
This problem is a linear algebra problem
Step One: Consider first statement
a) In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
What this means is that a matrix is that a matrix can have more than one reduced echelon form,according to the uniqueness of the reduced echelon form theorem,the reduced echelon form is always unique so since that is the case this gives given statement here is FALSE because it implies that one matrix can have more than one reduced echelon form but the uniqueness of reduced echelon form theorem says that the reduced echelon form of a matrix is unique and unique means that there is only one form of the reduced matrix so the statement is false
Step Two: Consider second statement
This statement states that the row reduction algorithm applies only to augmented matrices for a linear system.
This statement is FALSE because not only an augmented matrix can be reduced using the row reduction algorithm as quoted under the topic row reduction and echelon forms any matrix can be reduced using the row reduction algorithm not only the augmented matrix so an augmented matrix is one where let say for example you have a system
[tex]x_{1} + 2x_{2} =3\\2x_{1} + 4x_{2} = 2[/tex]
The coefficient matrix would be only the coefficients of the variable on the left hand side and that would be
[tex]\left[\begin{array}{ccc}1&2\\2&4\end{array}\right][/tex]
But the augmented includes the coefficient of the variables on the left hand side and the answers on the right side that is
[tex]\left[\begin{array}{ccc}1&2&3\\2&4&2\end{array}\right][/tex]
This statement say that the row reduction algorithm only applies to augmented matrix for a linear system but however again under the topic called row reduction and echelon form there you can see that any matrix can be reduced using the row reduction algorithm so it does not necessarily have to be augmented matrix it can also be coefficient matrix or any other type of matrix of linear system so the system is false
Step Three : Consider Third statement
This reads that a basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix
This statement is TRUE.From the definition of basic variable basic variable corresponds column that have leading ones so
Basic Variable → Column that have leading ones
and leading ones are also know pivot columns
For example let this matrix
[tex]\left[\begin{array}{ccc}1&0&3\\0&1&2\end{array}\right][/tex]
The first column has a leading one and the second column also has a leading one so basic variables are column with leading ones so column one and two are basic variables so a basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix and this is true
Step Four : Consider Fourth statement
This reads that finding a parametric description of the solution set of a linear system is the same as solving the system.So basically what it is saying is that solving a linear system is nothing but writing a parametric description of its solution set .This statement is true, the reason for that is because the solution set of a system of equations with infinitely many solutions can be represented by the free variables taking as parameters only as quoted under the topic parametric description of solution sets so the statement is true.
