true/false matrices: which of the following are true

Question

A. If 𝐴 is an 𝑛×𝑛 matrix and if the equation 𝐴𝑥=𝑏 does not have a solution for some 𝑛-vector 𝑏 in ℝ𝑛, then 𝐴 cannot have a pivot position in every row.

B. The matrix 𝐴 is 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 if the augmented matrix [ 𝐴 | 𝑏 ] has a pivot position in every row.

C. If the linear system whose augmented matrix is [ 𝐀1 𝐀2 𝐀3| 𝑏 ] has a unique solution, then the matrix 𝐴=[ 𝐀1 𝐀2 𝐀3 ] is 𝑛𝑜𝑛𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟.

D. Any linear combination of 𝑛-vectors in ℝ𝑛 can always be written in the form 𝐴𝑥 for a suitable matrix 𝐴 and vector 𝑥.

E. If the augmented matrix [ 𝐴 | 𝑏 ] does not have a pivot position in every row, then the matrix 𝐴 is 𝑛𝑜𝑛𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟.

F. If the equation 𝐴𝑥=𝑏 does not have a solution, then the matrix 𝐴 is 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟.

Answer 1

A. True. If [A|b] doesn't have a solution, then rref[A|b] contains a row of the form [0...0|1]. Note that A has no pivot in this row, since the pivot is in the augmented part.

B. False. Take A as the identify and b as the first standard basis vector e(1). Then [A|b] has a pivot in ever row, but A is not singular.

C. False. Take A to be a tall, skinny matrix whose columns all contain a pivot.

D. True. The vectors in the linear combination form the columns of A. The coefficients in the linear combination form the entries of x.

E. False. Note rref[A|b] contains a row of zeroes, hence so must rref[A], making A singular.

F. True. If Ax=b has no solution for some b, then A cannot have columns that span all of R(m), which means A is singular.

Answer 2

A. True. If an equation has no solution, we will have a row of zeros, so no pivot position in that row.

B. False. A pivot position in every row means the determinant is not zero which means it is non-singular (invertible).

C. True. If every column is a pivot column, then the determinant is not zero, hence it is non-singular.

D. True. Any linear combination of vectors can be written in the form Ax for a matrix A.

E. False. No pivot position in every row means that the determinant of A is zero, or A is singular.

F. True. If the equation Ax = b does not have a solution then we get a row of zeros. This means the determinant will be zero, hence the matrix is singular. A matrix is singular if and only if (iff) the determinant is zero.

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