1.2 Gaussian Elimination and Gauss-Jordan Elimination

Question 1

Define matrix

Answer 1

A rectangular array denoted by m x n. m and n are positive integers defining number of rows and columns respectively. Normally named using capital letters.

Question 2

entry

Answer 2

Number in a matrix. Denoted by aij, where index i is the row subscript and j is the column subscript

Question 3

real matrix

Answer 3

A matrix in which each entry is a real number

Question 4

Size of the matrix

Answer 4

Number of rows x number of columns

Question 5

Square matrix

Answer 5

A matrix with the same # of rows and columns

Question 6

A matrix with 4 rows and columns is ______ 4. The main diagonal entries of such a matrix would be ________.

Answer 6

square of order; a11, a22, a33, a44

Question 7

The matrix containing only the coefficients of the system

Answer 7

coefficient matrix

Question 8

The matrix derived from the coefficients and constant terms of a system of linear equations

Answer 8

augmented matrix

Question 9

Name and list the matrix equivalent of the three operations that produce equivalent systems of linear equations

Answer 9

elementary row operations:

1. Interchange two rows.
2. Multiply a row by a nonzero constant.
3. Add a multiple of a row to another row.

Question 10

List the properties of a matrix in row-echelon form

Answer 10

1. Any rows consisting entirely of zeros occur at the bottom of the matrix.
2. For each row that does not consist entirely of zeros, the first nonzero entry is (called a leading 1).
3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.

Question 11

A matrix in row-echelon form is in reduced row-echelon form when every column that has a leading 1 has zeros in every position ________its leading 1

Answer 11

above and below

Question 12

(T/F) Every matrix is row-equivalent to a matrix in row-echelon form

Answer 12

True

Question 13

Describe the three steps involved in Gaussian Elimination with back-substitution

Answer 13

1. Write the augmented matrix of the system of linear equations.
2. Use elementary row operations to rewrite the matrix in row-echelon form.
3. Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution.

Question 14

During Gaussian Elimination with back-substitution, you should operate _______ by _________ to obtain 0s in entries directly below leading 1s

Answer 14

left to right; columns

Question 15

If during Gaussian Elimination you obtain a row of all zeros except for the last entry, you should conclude _________

Answer 15

The system has no solution, and is inconsistent

Question 16

In Gauss-Jordan elimination, elementary row operations are performed until a _______ form is obtained

Answer 16

reduced row-echelon

Question 17

A homogenous system with more variables than equations has _______ solutions

Answer 17

infinitely many

Question 18

Explain why a homogenous system of linear equations will be consistent

Answer 18

A homogenous system will always be consistent because the zero solution (aka the trivial solution) will solve the system

Question 19

Systems of linear equations in which each of the constant terms is zero are ______

Answer 19

homogenous