Solving Systems of Equations with Gaussian and Gauss-Jordan Elimination

What is the first step in transforming a matrix into triangular form?

• Add the first row to the second row

• Swap the first and second rows

• Divide the first row by the leading coefficient

• Multiply the first row by -1 and add to the second row

In the context of matrix transformations, what does exchanging rows help achieve?

• Simplify calculations and avoid fractions

• Create a zero matrix

• Directly solve the system

• None of the above

What is the purpose of back substitution in solving matrices?

• To add one row to another

• To multiply rows by scalars

• To find the values of variables starting from the last row

• To convert the matrix to row echelon form

What defines a matrix in row echelon form?

• All entries above and below the leading ones are zero

• All entries are non-zero

• Each leading entry is to the right of the one above it

• All leading entries are one and all rows of zeros are at the bottom

Which of the following is a condition for a matrix to be in reduced row echelon form?

• Each leading entry must be zero

• Leading entries must shift left in successive rows

• All rows must consist entirely of zeros

• Every number above and below each leading entry must be zero

What operation is performed to create leading ones in Gaussian elimination?

• Row multiplication

• Row division

• Row addition

• Row swapping

What is achieved by multiplying a row by a scalar in Gaussian elimination?

• None of the above

• Eliminating a variable

• Creating a leading one

• Swapping two rows

How do you obtain zeros above each leading entry in Gauss-Jordan elimination?

• None of the above

• Multiply and subtract the row from all rows above it

• Divide the row by the leading entry

• Add the rows together

What indicates an inconsistent system when using Gaussian elimination?

• A row of zeros equals a non-zero

• All rows are zeros

• All entries are zeros

• No leading ones can be formed

What does it mean if a system of equations has infinitely many solutions?

• All variables are zero

• The solution set can be expressed parametrically

• The system's matrix is inconsistent

• The system has no solutions