QUIZ 3

Let A be a square matrix. Which of the following statements is not equivalent to "A is invertible"? The matrix map x → A x is one-to-one A has a leading entry in every row A has a pivot position in every row The transpose of A is invertible The matrix map x → A x is onto

A has a leading entry in every row

A square matrix A has determinant zero. Which of the following statements must be true? A x = 0 has only the trivial solution A has an inverse A x = b has a unique solution for every vector b A is row equivalent to I A has no inverse

A has no inverse

Let T be a linear map with standard matrix representation A. If T is invertible then its inverse also has standard matrix representation A.

F

A linear map is invertible if and only if its standard matrix representation is invertible

T

A square matrix is invertible if and only if its columns are linearly independent

T

Every linear map is invertible

T

The determinant of a 1 x 1 matrix is equal to its only entry

T

The sign of (- 1) i + j can be found by looking up the (i, j) entry of a sign matrix

T

Alice and Bob would like to compute the determinant of a 5 x 5 matrix by doing a cofactor expansion. Alice would like to expand along a row with four zeros. Bob would like to expand along a column with two zeros. There are no other zeros in the matrix. Whose approach to calculating the determinant is more efficient? It is impossible to determine The two approaches are equally efficient Bob's approach is more efficient Neither approach is more efficient than the other Alice's approach is more efficient

Alice's approach is more efficient

Which method can be used to compute the matrix product A -1 B ? Invert the augmented matrix [ A | B ] Apply Gauss-Jordan elimination to A and then multiply by B Transpose the augmented matrix [ A | B ] Apply Gauss-Jordan elimination to [ A | B ]

Apply Gauss-Jordan elimination to [ A | B ]

Which method can be used to compute the inverse of a square matrix A? Apply Gauss-Jordan elimination to [ A | I ] Invert the augmented matrix [ A | I ] Transpose the augmented matrix [ A | I ] Apply Gauss-Jordan elimination to A

Apply Gauss-Jordan elimination to [ A | I ]

Alice and Bob would like to compute the determinant of a 5 x 5 matrix by doing a cofactor expansion. Alice would like to expand along row 4 and Bob would like to expand along column 2. Whose approach to calculating the determinant is correct? Both approaches are correct Only Bob's approach is correct It is impossible to determine Neither approach is correct Only Alice's approach is correct

Both approaches are correct

A square matrix is invertible if and only if it has non-zero cofactors along row 1

F

A square matrix is invertible if and only if its columns span the range

F

All 2 x 2 matrices are invertible because the condition for not having an inverse can only be satisfied by 3 x 3 matrices or larger

F

Let A be a square matrix. Which of the following statements is not true? A and AT have the same determinant If A has a zero row then det (A) = 0 If det(A) = 0 then A must have either a zero row or a zero column If A has a zero column then det (A) = 0 If A is triangular then det (A) is equal to the product of the diagonal entries of A If A has two identical columns then det (A) = 0 If A has two identical rows then det (A) = 0

If det(A) = 0 then A must have either a zero row or a zero column

Let A = [aij] be a square matrix. The minor matrix Aij of aij is obtained by Removing row i and column j from A Keeping only row i and column j in A Computing (- 1) i + j det(A) Multiplying every entry of A by (- 1) i + j

Removing row i and column j from A

Which of the following methods cannot be used to solve A x = b? Solve the system with augmented matrix [ U | b ] where U is the reduced row echelon form of A Apply Gaussian elimination to [A | b], solve for the lead variables in terms of the free variables, and then apply back substitition Apply Gauss-Jordan elimination to [A | b] and then solve for the lead variables in terms of the free variables Reduce [ A | I ] to [ I | A - 1 ] and then compute x = A - 1 b

Solve the system with augmented matrix [ U | b ] where U is the reduced row echelon form of A

If a linear map is invertible, then its inverse is unique

T

Let A be a square matrix. Which of the following statements is not equivalent to "A is invertible"? A x = 0 has only the trivial solution A is row equivalent to I There is a matrix B such that B A = I and A B = I There is a matrix B such that B = 1 / A A x = b has a unique solution for every vector b

There is a matrix B such that B = 1 / A

Let A = [aij] be a square matrix. The minor of aij is det (Aij) where Aij is the minor matrix of aij (- 1) i + j det (Aij) where Aij is the minor matrix of aij det (Aij) where Aij is the major matrix of aij The matrix Aij obtained by removing row i and column j from A

det (Aij) where Aij is the minor matrix of aij

If A is a square matrix and E is an elementary matrix of type II then det (E A) = det (E) det (E A) = - det (A) det (E A) = det (I) where I is the identity matrix det (E A) = det (A)

det (E A) = - det (A)

If A is a square matrix and E is an elementary matrix of type I then det (E A) = det (I) where I is the identity matrix det (E A) = - det (A) det (E A) = det (A) det (E A) = c det (A) where Ri + c Rj → Ri det (E A) = det (E)

det (E A) = det (A)

If E is an elementary matrix of type II then det (E) = - 1 det (E) = 0 det (E) = E det (E) = 1

det (E) = - 1

If E is an elementary matrix of type I then det (E) = - 1 det (E) = 0 det (E) = E det (E) = 1

det (E) = 1

The determinant of a square matrix A is mistakenly computed using the cofactors from another row. The resulting sum is the determinant of AT the determinant of A - 1 nonsense an elementary row operation of type IV zero

zero