LINEAR ALGEBRA FINAL EXAM
QUESTIONS WITH COMPLETE
SOLUTIONS
A system of one linear equation in two variables is always consistent - Answer-True
A system of two linear equations in three variables is always consistent - Answer-False
If a linear system is consistent, then it has infinitely many solutions - Answer-False
A linear system can have exactly two solutions - Answer-False
Two systems of linear equations are equivalent when they have the same set - Answer-
True
A system of three linear equations in two variables is always inconsistent - Answer-
False
Every matrix is row-equivalent to a matrix in row-echelon form - Answer-True
If the row echelon form of the augmented matrix of a system of linear equations
contains the row [1,0,0,0,0] then the original system is inconsistent - Answer-False
A homogenous system of four linear equations in six variables has infinitely many
solutions - Answer-True. More variables than equations so infinite.
A homogenous system of four linear equations in four variables is always consistent -
Answer-True
The system Ax=b is consistent if and only if b can be expressed as a linear combination
of the columns of A, where the coefficients of the linear combination are a solution of
the system - Answer-True
Matrix addition is commutative - Answer-True
Matrix multiplication is associative - Answer-True
The transpose of the product of two metrics equals the product of their transposes that
is (AB)^T=A^TB^T - Answer-False
For any matrix C, the matrix CC^T is symmetric - Answer-True
Matrix multiplication is commutative - Answer-False
, If the matrices A,b,C satisfy AB=AC, then B=C - Answer-False
Every matrix A has an additive inverse - Answer-True
The transpose of the same of two matrices equals the sum of their transposes -
Answer-True
If the matrices A,B,C satisfy BA=CA and A is invertible, then B=C - Answer-True
The inverse of the product of two metrics is the product of their inverses; that is, (AB)^-
1=A^-1B^-1 - Answer-False
The inverse of the inverse of a nonsingular matrix A is equal to itself - Answer-True
If A is a square matrix, then the system of linear equations Ax=b has a unique solution -
Answer-False
The identity matrix is an elementary matrix - Answer-True
If E is an elementary matrix, then 2E is an elementary matrix - Answer-False
The inverse of an elementary matrix is an elementary matrix - Answer-True
The zero matrix is an elementary matrix - Answer-False
A square matrix is nonsingular when it can be written as the product of elementary
matrices - Answer-True
Ax=0 has only the trivial solution if and only if Ax=b has a unique solution for every nx1
column matrix b - Answer-False
Two vectors in R^n are equals if and only if their corresponding components are equal -
Answer-True
The vector -v is called the additive identity of the vector v - Answer-False
To subtract two vectors in R^n, subtract their corresponding components - Answer-True
The zero vector 0 in R^n is defined as the additive inverse of a vector - Answer-False
A vector space consists of four entries: a set of vectors, a set of scalars, and two
operations - Answer-True
The set of all integers with the standard operations is a vector space - Answer-False
QUESTIONS WITH COMPLETE
SOLUTIONS
A system of one linear equation in two variables is always consistent - Answer-True
A system of two linear equations in three variables is always consistent - Answer-False
If a linear system is consistent, then it has infinitely many solutions - Answer-False
A linear system can have exactly two solutions - Answer-False
Two systems of linear equations are equivalent when they have the same set - Answer-
True
A system of three linear equations in two variables is always inconsistent - Answer-
False
Every matrix is row-equivalent to a matrix in row-echelon form - Answer-True
If the row echelon form of the augmented matrix of a system of linear equations
contains the row [1,0,0,0,0] then the original system is inconsistent - Answer-False
A homogenous system of four linear equations in six variables has infinitely many
solutions - Answer-True. More variables than equations so infinite.
A homogenous system of four linear equations in four variables is always consistent -
Answer-True
The system Ax=b is consistent if and only if b can be expressed as a linear combination
of the columns of A, where the coefficients of the linear combination are a solution of
the system - Answer-True
Matrix addition is commutative - Answer-True
Matrix multiplication is associative - Answer-True
The transpose of the product of two metrics equals the product of their transposes that
is (AB)^T=A^TB^T - Answer-False
For any matrix C, the matrix CC^T is symmetric - Answer-True
Matrix multiplication is commutative - Answer-False
, If the matrices A,b,C satisfy AB=AC, then B=C - Answer-False
Every matrix A has an additive inverse - Answer-True
The transpose of the same of two matrices equals the sum of their transposes -
Answer-True
If the matrices A,B,C satisfy BA=CA and A is invertible, then B=C - Answer-True
The inverse of the product of two metrics is the product of their inverses; that is, (AB)^-
1=A^-1B^-1 - Answer-False
The inverse of the inverse of a nonsingular matrix A is equal to itself - Answer-True
If A is a square matrix, then the system of linear equations Ax=b has a unique solution -
Answer-False
The identity matrix is an elementary matrix - Answer-True
If E is an elementary matrix, then 2E is an elementary matrix - Answer-False
The inverse of an elementary matrix is an elementary matrix - Answer-True
The zero matrix is an elementary matrix - Answer-False
A square matrix is nonsingular when it can be written as the product of elementary
matrices - Answer-True
Ax=0 has only the trivial solution if and only if Ax=b has a unique solution for every nx1
column matrix b - Answer-False
Two vectors in R^n are equals if and only if their corresponding components are equal -
Answer-True
The vector -v is called the additive identity of the vector v - Answer-False
To subtract two vectors in R^n, subtract their corresponding components - Answer-True
The zero vector 0 in R^n is defined as the additive inverse of a vector - Answer-False
A vector space consists of four entries: a set of vectors, a set of scalars, and two
operations - Answer-True
The set of all integers with the standard operations is a vector space - Answer-False
