Test 1

Question 1

Every Matrix is ROW EQUIVALENT to a UNIQUE matrix in ECHELON form

Answer 1

FALSE

Question 2

If an augmented matrix [A b]is TRANSFORMED into [C d] by ELEMENTARY row operations, then the equations Ax=b and Cx=d have EXACTLY THE SAME solution set

Answer 2

TRUE

Question 3

If a system Ax=b has MORE THAN ONE solution, then SO DOES the system Ax= 0

Answer 3

TRUE

Question 4

If Matrices A and B are ROW EQUIVALENT, they have the SAME REDUCED ECHELON form.

Answer 4

TRUE

Question 5

If A is an (m x n) matrix and the equation Ax=b is CONSISTENT for every b in R^m, then A has m PIVOT COLUMNS.

Answer 5

TRUE

Question 6

If (3x3) matrices A and B EACH HAVE THREE PIVOT positions, then A CAN be TRANSFORMED into B by elementary row operation

Answer 6

TRUE

Question 7

If {u, v, w} is Linearly INDEPENDENT, then u,v and w are NOT in R^2

Answer 7

TRUE

Question 8

If u,v and w are NON-ZERO vectors in R^2, then w is a Linear COMBINATION of u an v

Answer 8

FALSE

Question 9

If A nd B are (m x n), then both (AB)’transpose and (A)’transpose.B are defined

Answer 9

TRUE

Question 10

If AB=C and C has 2 columns, then A has 2 columns

Answer 10

FALSE

Question 11

Left-multiplying a matrix B by a DIAGONAL matrix A, with NON-ZERO entries on the diagonal, SCALES the ROW of B

Answer 11

TRUE

Question 12

If A and B are (n x n), then (A+B)(A-B) = A^2 - B^2.

Answer 12

FALSE

Question 13

the Transpose of an elementary matrix is an elementary matrix

Answer 13

TRUE

Question 14

An elementary matrix must be square

Answer 14

TRUE

Question 15

Every square matrix is a product of elementary matrices

Answer 15

FALSE

Question 16

If the equation Ax= 0 has ONLY the TRIVIAL solution, then A is ROW EQUIVALENT to the (n x n) IDENTITY matrix

Answer 16

TRUE

Question 17

If there is an (n x n) matrix D such that AD = Identity, then there is also an (n x n) matrix C such that CA = Identity

Answer 17

TRUE

Question 18

If the columns of A span R^n, then the columns are Linearly INDEPENDENT

Answer 18

FALSE

Question 19

If A is an (n x n) matrix, then the equation Ax=b has AT LEAST ONE SOLUTION for EACH b in R^n

Answer 19

FALSE

Question 20

If the equation Ax= 0 has NON-TRIVIAL solution, then A has fewer than n pivot positions

Answer 20

TRUE

Question 21

If A’transpose is not invertible, then A is not invertible

Answer 21

TRUE

Question 22

If A and B are (n x n) and invertible, then A^-1.B^-1 is the inverse of AB

Answer 22

FALSE - must be B^-1.A^-1

Question 23

An upper triangular matrix is invertible IFF all elements on diagonal are non-zero

Answer 23

TRUE

Question 24

INVERTABLE MATRIX THEOREM of Square Matrix A (n x n)
1 - A is an invertible Matrix
2 - A is row equivalent to the n x n identity matrix
3 - A has n pivot positions
4 - Ax=0 ONLY TRIVIAL solution
5 - Columns of A form a linearly INDEPENDENT set
6 - Linear Transformation x -> xA is one-to-one
7 - Ax=b has at least one solution for each b in R^n
8 - Columns of A span R^n
9 - Linear Transformation x -> Ax map R^n onto R^n
10 - There exists an n x n matrix C such that CA = Identity
11 - There exists an n x n matrix D such that AD = Identity
12 - A’transpose is an invertible matrix

Answer 24

ALL Must be TRUE

Question 26

INVERTIBALE MATRIX THEOREM - for Square matrix A (n x n)
1 - A is an invertible matrix
2 - A is row equivalent to the n x n identity matrix
3 - A has n pivot positions
4 - The equation Ax=0 has ONLY TRIVIAL solution
5 - The columns of A form a linearly independent set
6 - The linear transformation x -> Ax is one-to-one
7 - The equation Ax=b has at least one solution for each b in R^n
8 - The columns of A span R^n
9 - The linear transformation x - >Ax maps R^n to R^n
10 - There exists an n x n matrix C such CA = Identity
11 - There exists an n x n matrix D such that AD = Identity
12 - A’transpose is an invertible matrix

Answer 26

ALL must be TRUE

Question 27

If A and B are 2x2 with columns a1, a2, and b1, b2 respectively, then AB = [a1b1 a2b2]

Answer 27

FALSE - result is 2x2

Question 28

If A and B are 3x3 and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3]

Answer 28

FALSE - result is 3 x 3

Question 29

Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A

Answer 29

TRUE

Question 30

AB + AC = A(B+C)

Answer 30

TRUE

Question 31

A^T + B^T = (A+B)^T

Answer 31

TRUE

Question 32

(AB) C = (AC) B

Answer 32

FALSE - order must sustain

Question 33

(AB)^T = A^T B^T

Answer 33

TRUE

Question 34

The Transpose of a product of matrices equals the product of their transposes in the same order

Answer 34

FALSE - must be reverse order