True False Linear Algebra

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(T) Every invertible matrix is diagonalizable

False

(T) If a matrix A is multiplied by a scalar c, the determinant of the resulting matrix is c*det(A)

False

(T) If det(A) = 2 and det(B) = 3 then det(A+B) = 5

False

A n x n is diagonal if and only if n distinct values

False

Every n x n matrix has distinct e values

False

Every square matrix has real Eigen values

False

Every square matrix is diagonal

False

Every triangular matrix is dragon

False

For any vector in a in R(3), we have ||a x b|| = ||a^2||

False

If A is n x n diagonal matrix there is a unique D that is similar to A

False

If two rows and also two columns of a square matrix A are interchanged, the determinant changes sign

False

If u + v lies in a subspace W of a vector space V, then both u and v are in W

False

If λ is an e-value ... then it's one for A+cI

False

In order for the determinant of a 3 x 3 matrix to be zero, two rows must be parallel

False

Matrix Multiplication is a vector space operation on the set of all square matrices

False

Square matrix is non-singular if and only if it's determined is positive

False

The determinant det(A) is defined for any matrix A

False

The determinant of a 2X2 matrix is a vector

False

The determinant of a 3 x 3 matrix is zero if the points of R(3) given by the rows of the matrix lie in a plane

False

The determinant of a square matrix is the product of all the entries on its main diagonal

False

The formula for A-1 is practical

False

The parallelogram in R(2) determined by non-zero vectors a and b is a square if and only if a dot b = 0

False

The product of a square matrix and it's adjoins is the identity matrix

False

There can be only one e vector Associated with an E value of a linear transformation

False

If A^2 = A, then A = I or 0

False, A = [1,0,1,0]

If AC = BC, then A = B

False, C = 0

Every subset of three non zero vectors in R^2 spans R^2

False, could be dependent set

(T) If V is in a eigenvector of a matrix, then V is a eigenvector of A +cI for all scalars c

True

(T) If an n x n matrix is multiplied by a scalar c, the determinant of the resulting matrix is c^n*det(A)

True

(T) If two rows of a 3 x 3 matrix are interchanged, the sign of the determinant is changed

True

A homogeneous square linear system has a non trivial solution if and only if the determinant of its coefficient matrix is zero

True

A linear transformation having an M x N matrix as a standard matrix representation maps R^n onto R^M

True

Every Subset of four vectors in R^3 is dependent

True

Every elementary Matrix is invertible

True

Every n x n has n e values that may not be distinct

True

Every n x n real symmetrical matrix is real diagonal

True

For every square matrix A, we have det(AA^T) = det(A^tA) = det(A)^2

True

If A = B, then AC = BC

True

If A and B are similar and square, det(A) = det(B)

True

If A and B are similar both are diagonal if the other is

True

If V is a vector space of dimension n, then V is isomorphic to R^n

True

If a square matrix has n real e values it is diagnolizable

True

If det(A) = 2 and det(B) = 3 then det(AB) = 6

True

If the angle between vectors a and b in R(3) IS pi/4 then ||a x b|| = |a dot b|

True

If v is an e-vector if an invertibile matrix A, then cv is an e-vector of A-1

True

The box in R(3) determined by vectors a,b, and c is a cube if and only if all dot products equal zero

True

The column vectors of an n x n matrix are independent if and only if the determinant of the matrix is non zero

True

The determinant det(A) is defined for each square matrix A

True

The determinant of a 3 x 3 matrix is zero if the points in R(3) given by the row of the matrix lie in a plane through the origin

True

The determinant of a 3 x 3 matrix is zero if two rows of the matrix are parallel vectors in R(3)

True

The determinant of a lower triangular square matrix is the product of all entries on its main title

True

The determinant of a square matrix is a scalar

True

The determinant of an elementary matrix is non zero

True

The determinant of an upper triangular square matrix is the product of all entries on its main diagonal

True

The number of independent row vectors in a matrix is the same as the number of independent column vectors

True

The product of a square matrix and it's adjoint matrix is equal to a scalar times the identity matrix

True

The transpose if the adjoint matrix is the matrix of cofactors

True

The vector space P8 is isomorphic to R^9

True

There can be only one e value associated with e vector of a transformation

True


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