True/False
If a matrix is in reduced row echelon form, then the first nonzero entry in each row is a 1 and has 0s below it.
True
If the characteristic polynomial of a 2×2 matrix is λ2−5λ+6 then the determinant is 6.
True
If the columns of an m×n matrix A span Rm, then the equation Ax=b is consistent for each b in Rm.
True
If the solution to a system of linear equations is given by (4−2z,−3+z,z), then (4,−3,0) is a solution to the system.
True
If there is a basis of Rn consisting of eigenvectors of A|, then A is diagonalizable|.
True
If two columns of a matrix are the same, then the determinant of that matrix is zero.
True
If y is in the subspace W and its orthogonal complement W⊥, then y must be the zero vector.
True
A row replacement operation does not affect the determinant of a matrix.
True
The eigenvalues of A| are the entries on its main diagonal.
False
Define diagonalizable.
A matrix is diagonalizable if it is similar to a diagonal matrix.
Define eigenvector.
An eigenvector for a matrix A is a nonzero vector v so that Av is equal to a multiple of v.
The eigenvalues of a projection matrix are −1 and 1.
False
The equation Ax=b is consistent if the augmented matrix [A|b] has a pivot position in every row.
False
If an n×n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.
False
If the columns of A are linearly independent, then det A=0.
False
det (A+B) = det A + det B
False
If the equation Ax=0 has the trivial solution, then the columns of A span Rn. (Yes/No/Maybe)
Maybe
A square matrix with two identical columns can be invertible. (Yes/No/Maybe)
No
If the linear transformation TA(x)=Ax is one-to-one, then the columns of A form a linearly dependent set. (Yes/No/Maybe)
No
Suppose A is an 5×5 matrix. If A has three pivots, then ColA is a two-dimensional plane. (Yes/No/Maybe)
No
Suppose A is an 5×5 matrix. If A has two pivots, then the dimension of NulA is 2. (Yes/No/Maybe)
No
Suppose A is an 5×5 matrix. If rankA=4 , then the columns of A form a basis of R^5. (Yes/No/Maybe)
No
Let V be the subset of R^3 consisting of the vectors ⎡⎣a b c⎤⎦ with abc=0. V contains the zero vector.
True
Let V be the subset of R^3 consisting of the vectors ⎡⎣a b c⎤⎦ with abc=0. V is closed under scalar multiplication, meaning that if u is in V and c is a real number then cu is in V.
True
The absolute value of the determinant of A equals the volume of the parallelepiped determined by the columns of A.
True
If A is invertible, then the equation Ax=b has exactly one solution for all b in R^n. (Yes/No/Maybe)
Yes
A 5×5| real matrix has an even number of real eigenvalues.
False
A least-squares solution of Ax=b is a vector x^ such that ||b−Ax||≤||b−Ax^|| for all x in Rn.
False
A set of three vectors in R4 can span all of R4.
False
A| is diagonalizable if and only if A| has n eigenvalues, counting multiplicity.
False
For any matrices A and B, if the product AB is defined, then BA is also defined.
False
If A is a 5×4 matrix, and B is a 4×3 matrix, then the entry of AB in the 3rd row / 4th column is obtained by multiplying the 3rd column of A by the 4th row of B.
False
If A| is diagonalizable, then A| is invertible.
False
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for the column space of A.
False
If S is a set of linearly dependent vectors, then every vector in S can be written as a linear combination of the other vectors in S.
False
If a linear system has four equations and seven variables, then it must have infinitely many solutions.
False
If a set S of vectors contains fewer vectors than there are entries in the vectors, then the set must be linearly independent.
False
If an augmented matrix in reduced row echelon form has 2 rows and 3 columns (to the left of the vertical bar), then the corresponding linear system has infinitely many solutions.
False
If det A is zero, then two columns of A must be the same, or all of the elements in a row or column of A are zero.
False
If x is a nontrivial solution of Ax=0, then every entry of x is nonzero.
False
If {v1,v2,v3} is an orthogonal basis for W, then multiplying v3 by a scalar c gives a new orthogonal basis {v1,v2,cv3}.
False
Let V be the subset of R^3 consisting of the vectors ⎡⎣a b c⎤⎦ with abc=0. V is a subspace of R^3 .
False
Let V be the subset of R^3 consisting of the vectors ⎡⎣a b c⎤⎦ with abc=0. V is closed under vector addition, meaning that if u and v are in V then u+v is in V.
False
Matrices with the same eigenvalues are similar matrices.
False
Row operations on a matrix do not change its eigenvalues.
False
Span{a1,a2} contains only the line through a1 and the origin, and the line through the a2 and the origin.
False
Suppose A and B are invertible matrices. (A+B)^2=A^2+B^2+2AB.
False
Suppose A and B are invertible matrices. (AB)^−1=A^−1B^−1.
False
Suppose A and B are invertible matrices. A+B is invertible.
False
The columns of matrix A are linearly independent if equation Ax=0 has the trivial solution.
False
The determinant of a triangular matrix is the sum of the entries of the main diagonal.
False
The homogeneous system Ax=0 has the trivial solution if and only if the system has at least one free variable.
False
The null space of an m×n matrix is a subspace of R^m.
False
There are exactly three vectors in Span{a1,a2,a3}.
False
There exists a real 2×2 matrix with the eigenvalues i and 2i.
False
λ| is an eigenvalue of matrix A| if A−λI has linearly independent columns.
False
A determinant of an n×n matrix can be defined as a sum of determinants of (n−1)×(n−1) submatrices.
True
A homogeneous system is always consistent.
True
A matrix that is similar to the identity matrix is equal to the identity matrix.
True
A number c| is an eigenvalue of A| if and only if (A - cI)v = 0 has a nontrivial solution.
True
Asking whether the linear system corresponding to an augmented matrix [a1a2a3|b] has a solution amounts to asking whether b is in Span{a1,a2,a3}.
True
A| is invertible if and only 0 is not an eigenvalue of A|.
True
Every real 3×3| matrix must have a real eigenvalue.
True
For any matrix A, there exists a matrix B so that A+B=0.
True
For any matrix A, we have the equality 2A+3A=5A.
True
If A is a projection matrix, then A^2=A.
True
If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in Rm, then A cannot have a pivot position in every row.
True
If A is an m×n matrix then A^TA and AA^T are both defined.
True
If A is an m×n matrix whose columns do not span Rm, then the equation Ax=b is inconsistent for some b in Rm.
True
If A is diagonalizable, then A^2 is also diagonalizable.
True
If A=QR, where Q has orthonormal columns, then R=Q^TA.
True
If A| is nxn and A| has n distinct eigenvalues, then the eigenvectors of A| are linearly independent.
True
If W=Span{x1,x2,x3} with {x1,x2,x3} linearly independent, and if {v1,v2,v3} is an orthogonal set in W, then {v1,v2,v3} is a basis for W.
True
If b is in the column space of A, then every solution of Ax=b is a least-squares solution.
True
If the equation Ax=b is consistent, then b is in the span of the columns of A.
True
If v| is an eigenvector of A|, then cv| is also an eigenvector of A| for any number c that doesn't equal 0.
True
If x is not in a subspace W, then x−projWx is not zero.
True
If y is in the subspace W then the orthogonal projection of y onto W is y.
True
Real eigenvalues of a real matrix always correspond to real eigenvectors.
True
Suppose A and B are invertible matrices. (In−A)(In+A)=In−A^2.
True
Suppose A and B are invertible matrices. A^7 is invertible.
True
The (i,j) minor of a matrix A is the matrix Aij obtained by deleting row i and column j from A.
True
The Gram-Schmidt process produces from a linearly independent set {x1,...,xp} an orthogonal set {v1,...,vp} with the property that for each k, the vectors v1,...,vk span the same subspace as that spanned by x1,...,xk.
True
The cofactor expansion of det A along the first row of A is equal to the cofactor expansion of det A along any other row.
True
The column space of an m×n matrix is a subspace of R^m.
True
The columns of a matrix with dimensions m×n, where m<n, must be linearly dependent.
True
The columns of an invertible n×n matrix form a basis for R^n.
True
The equation Ax=b is homogenous if the zero vector is a solution.
True
The first entry in the product Ax is a sum of products.
True
The general least-squares problem is to find an x that makes Ax as close as possible to b.
True
The least-squares solution of Ax=b is the point in the column space of A closest to b.
True
The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of R^n.
True
The solution set of Ax=b is obtained by translating the solution set of Ax=0.
True
The solution set of a linear system whose augmented matrix is [ a1 a2 a3 |b ] is the same as the solution set of Ax=b, if A=[ a1 a2 a3 ].
True
The solution set of the linear system whose augmented matrix [a1a2a3|b] is the same as the solution set of the equation x1a1+x2a2+a3x3=b.
True
The vector b is a linear combination of the columns of matrix A if and only if Ax=b has at least one solution.
True
There are exactly three vectors in the set {a1,a2,a3}.
True
Two vectors are linearly dependent if and only if they are colinear.
True
If A^T is row equivalent to the n×n identity matrix, then the columns of A span R^n. (Yes/No/Maybe)
Yes
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot points. (Yes/No/Maybe)
Yes
If the linear transformation TA(x)=Ax is onto, then it is also one-to-one. (Yes/No/Maybe)
Yes
If the transpose of A is not invertible, then A is also not invertible. (Yes/No/Maybe)
Yes
Suppose A is an 5×5 matrix. If Ax=0 has only the trivial solution, then ColA=R^5. (Yes/No/Maybe)
Yes
The product of any two invertible matrices is invertible. (Yes/No/Maybe)
Yes