Systems of Linear Equations - short

The product of two elementary matrices of the same size must be an elementary matrix.

• True

• False

Answer explanation

False; not necessarily

Every elementary matrix is invertible.

• True

• False

If A and B are row equivalent, and if B and C are row equivalent, then A and C are row equivalent.

• True

• False

If A is an nxn matrix that is not invertible then the linear system Ax=0 has infinitely many solutions.

• True

• False

If A is an nxn matrix that is not invertible then the matrix obtained by interchanging the rows of A cannot be invertible.

• True

• False

If A is invertible and a multiple of the first row of A is added to the second row, then the resulting matrix is invertible.

• True

• False

An expression of an invertible matrix A as a product of elementary matrices is unique.

• True

• False

If A is a square matrix, and if the linear system Ax=b has a unique solution, then the linear system Ax=c also must have a unique solution.

• True

• False

If A and B are row equivalent matrices, then the linear systems Ax=0 and Bx=0 have the same solution set.

• True

• False

Answer explanation

Elementary row operations do not change the solution

Let A be an nxn matrix and S is an n x n invertible matrix. If x is a solution to the system (S^-1AS)x=b, then Sx is a solution to the system Ay=Sb.

• True

• False