Matrices Test 2

det(A+B)=det A+ det B

False

(ABC)^T=C^T*A^T*B^T

False, (ABC)^T=C^T*B^T*A^T

If A and B are 3X3 matrices with and B=[b1 b2 b3], then AB= [Ab1+Ab2+Ab3].

False, AB would be AB=[Ab1 Ab2 Ab3] without the addition.

If A and B are 2X2 matrices with columns a1, a2, and b1, b2, respectively, then AB= [a1b1 a2b2].

False, AB would be a 2X2 matrix as well and would be equal to...

If A and B are nXn and invertible, then A^-1*B^-1 is the inverse of AB.

False, B^-1*A^-1 is the inverse of AB.

A linearly independent set in a subspace H is basis for H.

False, a linearly independent set in a subspace H spanned by the set is a basis for H.

A vector is an arrow in 3 dimensional space.

False, a vector may have any number of dimensions, not just 3.

det A^T=(-)det A

False, det A^T=det A

The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U.

False, echelon form does not show invertibility and determinant would be zero in this case.

If ab-cd is not equal to 0, then A is invertible.

False, for A to be invertible ad-bc cannot equal 0.

A basis is a spanning set that is as large as possible.

False, for the space R^n there exist more spanning sets then the basis. But every spanning set need not be a basis because there is a case that the set fails to be LI.

The null space of an mXn matrix is in R^m.

False, it is in R^n.

The determinant of a triangular matrix is the sum of the entries on the main diagonal.

False, it is the product of the entities of the diagonal of A.

A single vector by itself is LD.

False, it would be LD by itself only if it is a zero vector.

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

False, it would equal the product of their transposes in reverse order.

The standard method for producing a spanning set for Nul A, sometimes fails to produce a basis for Nul A.

False, our method always produces a linearly independent set when Nul A contains nonzero vectors.

The cofactor expansion of set A down a column is the negative of the cofactor expansion along a row.

False, the cofactor expansion of set A down a column is the the same as the cofactor expansion along a row.

The (i,j)-cofactor of a matrix A is the matrix A(ij) obtained by deleting from A its ith row and nth column.

False, the cofactor is the determinant of the matrix A(ij) multiplied by -1^(i+j).

If A is invertible, then elementary row operations that reduce A to the identity I also reduce A^-1 to I.

False, the elementary row operations that reduces A to I also transforms I into A^-1.

If B is an echelon form of a matrix, then the pivot columns form a basis for Col A.

False, the pivot columns of a matrix A form a basis for Col A.

A subset H of a vector space V is a subspace of V if the zero vector is in H.

False, the presence of a zero vector in H is only one of the three properties required for H to be a subspace of V.

A product of invertible nXn matrices is invertible, and inverse of the product is the product of their inverses in the same order.

False, the product matrix is invertible but the products of the inverse should be in reverse order.

If det A is zero, then two rows or two columns are the same, or a row or column is zero.

False, the set A can be equal to 0 and have no two rows or two columns that are the same or a column not equal to zero.

If f is a function in the vector space V of all real-valued functions on R and if f(t)=0 for some t, then f is the zero vector in V.

False, the zero vector has the property 0+v=v+0=v but f(t)+v=v+f(t)=v may not be true for all values of t.

R^2 is a subspace of R^3.

False, there are only 2 entries in R^2 and R^3 requires 3 for a subspace.

In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.

False, when a matrix is row reduced using row ops it still has the same solution.

The determinant of A is the product of the diagonal entries in A.

False, when a matrix is triangular then its determinant is the product of its entries on the main diagonal but for other matrices this is not possible.

A basis is a linearly independent set that is as large as possible.

True

A null space is a vector space.

True

A row replacement operation does not affect the determinant of a matrix.

True

A subset H of a vector space V is a subspace of V if the following conditions are satisfied; (i) the zero vector of V is in H, (ii) u, v, and u+v are in H, and (iii) c is a scalar and cu is in H.

True

A subspace is also a vector space.

True

A vector is any element of a vector space.

True

A vector space is also a subspace.

True

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

True

An nXn determinant is defined by determinants of (n-1)X(n-1) sub matrices.

True

Analog signals are used in the major control systems for the space shuttle, mentioned in the introduction of the chapter.

True

Col A is the set of all solutions of Ax=b.

True

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

True

Each elementary matrix is invertible.

True

If A can be row reduced to the identity matrix, then A must be invertible.

True

If A is an invertible nXn matrix, then the equation Ax=b is consistent for each b in R^n.

True

If A is an nXn matrix and Ax=e(j) is consistent for every j in {1,2,3,...,n}, then A is invertible.

True

If A is an nXn matrix, then (A^2)^T=(A^T)^2

True

If A is invertible, then the inverse of A^-1 is A itself.

True

If H=Span {b1, . . . , bp}, the {b1, . . . ,bp} is a basis for H.

True

If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.

True

If the columns of A are LD, then det A=0.

True

If the equation Ax=b is consistent, then Col A is in R^m.

True

If two row interchanges are made in succession, then the new determinant equals the old determinant.

True

If u is a vector in a vector space V, then (-1)u is the same as the negative of u.

True

In order for a matrix B to be the inverse of A, the equations AB=I and BA=I must both be true.

True

Nul A is the kernel of the mapping A to Ax.

True

The column space of A is the range of the mapping x to Ax.

True

The column space of an mXn matrix is in R^m.

True

The columns of an invertible nXn matrix form a basis for R^n.

True

The first row of AB is the first row of A multiplied on the right by B.

True

The kernel of a linear transformation is a vector space.

True

The null space of A is the solution set of the equations Ax=0.

True

The range of a linear transformation is a vector space.

True

The set of all solutions of a homogenous linear differential equation is the kernel of a linear transformation.

True

The transpose of a sum of matrices equals the sum of their transposes.

True

Col A is the set of all vectors that can be written as Ax for some x.

False, Col A is the set of column vectors who have pivots if Ax=b is consistent. Col A is not the set of all solutions.

AB+AC=A(B+C)

True