Multivariable Calc Midterm – Questions With
Complete Solutions
If and if A is invertible, then A is always square. ✔️Ans - True
Row-equivalent linear systems can have different sets of solutions provided
the elements are similar. ✔️Ans - True
If AB = C and C has 2 columns, then A has 2 columns. ✔️Ans - False
The transpose of a product of matrices equals the product of their transposes
in the same order. ✔️Ans - False
is homogeneous if the zero vector is a solution. ✔️Ans - True
A vector of norm 1 is called a normal vector. ✔️Ans - False
If A and B are (m x n), then both (AB)^T and (A^T)B are defined. ✔️Ans -
True
If AB = I, then A is invertible. ✔️Ans - False
The rank of a matrix A is the minimum number of linearly independent row
vectors of A. ✔️Ans - True
Quiz 1, question 10 picture
Matrices A and B are row equivalent. Choose one mathematical statement that
can be made concerning the properties of the matrix A. ✔️Ans - -The span
of the columns of A form an incomplete set because there are no free
variables.
-The span of the columns of A intersect each other when plotted on the same
graph.
-All of the above.
-None of the above.
The vector space R^n consisting of all vectors with n components (i.e n real
numbers) has dimension n+1. ✔️Ans - False
, If an augmented matrix [a b] is transformed into [c d] by elementary row
operations, then the equation ax=b and cx = d have exactly the same solution
sets. ✔️Ans - True
Every matrix is row equivalent to a unique matrix in reduced echelon form.
✔️Ans - True
For a given matrix, A, the solution set of the homogeneous system
Ax=0
is not a vector space. ✔️Ans - False
The interchange of two rows in a matrix leaves the values of a determinant
unaltered. ✔️Ans - False
If a system of linear equations have two different solutions, they must have
infinitely many solutions. ✔️Ans - True
Matrix multiplication is not generally commutative. ✔️Ans - True
If a system of linear equations has no free variables, then it has a unique
solution or no solution. ✔️Ans - True
A homogeneous equation is always consistent. ✔️Ans - True
The row space and the column space of a matrix A have the same dimension,
equal to rank A. ✔️Ans - True
A linear transformation is not a function. ✔️Ans - False
When the eigenvalues of a skew-symmetric matrix are less than zero, it is
undefined. ✔️Ans - True
If the characteristic polynomial has degree n, then the sum of all the algebraic
multiplicities must equal n. ✔️Ans - True
Eigenvectors of an nxn matrix A forms a basis for R^n. ✔️Ans - False
An nxn matrix has at least one and at most n numerically different
eigenvalues. ✔️Ans - True
Complete Solutions
If and if A is invertible, then A is always square. ✔️Ans - True
Row-equivalent linear systems can have different sets of solutions provided
the elements are similar. ✔️Ans - True
If AB = C and C has 2 columns, then A has 2 columns. ✔️Ans - False
The transpose of a product of matrices equals the product of their transposes
in the same order. ✔️Ans - False
is homogeneous if the zero vector is a solution. ✔️Ans - True
A vector of norm 1 is called a normal vector. ✔️Ans - False
If A and B are (m x n), then both (AB)^T and (A^T)B are defined. ✔️Ans -
True
If AB = I, then A is invertible. ✔️Ans - False
The rank of a matrix A is the minimum number of linearly independent row
vectors of A. ✔️Ans - True
Quiz 1, question 10 picture
Matrices A and B are row equivalent. Choose one mathematical statement that
can be made concerning the properties of the matrix A. ✔️Ans - -The span
of the columns of A form an incomplete set because there are no free
variables.
-The span of the columns of A intersect each other when plotted on the same
graph.
-All of the above.
-None of the above.
The vector space R^n consisting of all vectors with n components (i.e n real
numbers) has dimension n+1. ✔️Ans - False
, If an augmented matrix [a b] is transformed into [c d] by elementary row
operations, then the equation ax=b and cx = d have exactly the same solution
sets. ✔️Ans - True
Every matrix is row equivalent to a unique matrix in reduced echelon form.
✔️Ans - True
For a given matrix, A, the solution set of the homogeneous system
Ax=0
is not a vector space. ✔️Ans - False
The interchange of two rows in a matrix leaves the values of a determinant
unaltered. ✔️Ans - False
If a system of linear equations have two different solutions, they must have
infinitely many solutions. ✔️Ans - True
Matrix multiplication is not generally commutative. ✔️Ans - True
If a system of linear equations has no free variables, then it has a unique
solution or no solution. ✔️Ans - True
A homogeneous equation is always consistent. ✔️Ans - True
The row space and the column space of a matrix A have the same dimension,
equal to rank A. ✔️Ans - True
A linear transformation is not a function. ✔️Ans - False
When the eigenvalues of a skew-symmetric matrix are less than zero, it is
undefined. ✔️Ans - True
If the characteristic polynomial has degree n, then the sum of all the algebraic
multiplicities must equal n. ✔️Ans - True
Eigenvectors of an nxn matrix A forms a basis for R^n. ✔️Ans - False
An nxn matrix has at least one and at most n numerically different
eigenvalues. ✔️Ans - True
