Chapter 2 T/F - Linear Algebra

If AB=C and C has 2 columns, then A had 2 columns.

False, B must have 2 columns. A has as many columns as B has rows.

If AC=0, then either A=0 or C=0.

False.

If A and B are n x n, then (A+B)(A-B) + A^2 - B^2.

False. (A + B)(A - B) = A^2 - AB + BA - B^2. This equals A2 - B2 if and only if A commutes with B.

If AB=I, then A is invertible.

False. A must be square in order to conclude from the equation AB = I that A is invertible.

If A and B are square and invertible, then AB is invertible, and (AB)^-1 = A^(-1)B^(-1).

False. AB is invertible, but (AB)^-1= B^(-1)A^(-1), and this product is not always equal to A^(-1)B^(-1).

Every square matrix is a product of elementary matrices.

False. Elementary matrices are invertible, so a product of such matrices is invertible. But not every square matrix is invertible.

If BC+BD, then C = D.

False. Take the zero matrix for B. Or, construct a matrix B such that the equation Bx = 0 has nontrivial solutions, and construct C and D so that C ≠ D and the columns of C - D satisfy the equation Bx = 0. Then B(C - D) = 0 and BC = BD.

If A is invertible and if r doesn't equal 0, then (rA)^-1 = rA^-1.

False. The correct equation is (rA)^(-1) = r^-1A^-1, because(rA)(r^-1A^-1) = (rr^-1)(AA^-1) = 1⋅I = I.

if A is a 3 x 3 matrix and the equation Ax = [1 0 0] (vertically), then A is invertible

True

If A and B are m x n then both AB^T and A^TB are defined.

True, if A and B are m×n matrices, then B^T has as many rows as A has columns, so AB^T is defined. Also, A^TB is defined because A^T has m columns and B has m rows.

An elementary matrix must be square.

True. An n×n elementary matrix is obtained by a row operation on In.

An elementary n x n matrix has either n or n+1 nonzero entries.

True. An n×n replacement matrix has n + 1 nonzero entries. The n×n scale and interchange matrices have n nonzero entries.

If AB=BA and if A is invertible, then A^(-1)B = BA^(-1).

True. Given AB = BA, left-multiply by A^(-1) to get B = A^(-1)BA, and then right-multiply by A^(-1) to obtain BA^(-1)= A^(-1)B.

If A is a 3x3 matrix with three pivot positions, there exist elementary matrices E1,..,Ep such that Ep...E1A = I.

True. If A is 3×3 with three pivot positions, then A is row equivalent to I3.

Left-multiplying a matrix B by a diagonal matrix A ,with nonzero entries on the diagonal, scales the rows of B

True. The ith row of A has the form (0, ..., di, ..., 0). So the ith row of AB is (0, ..., di, ..., 0)B, which is di times the ith row of B.

The transpose of an elementary matrix is an elementary matrix.

True. The transpose of an elementary matrix is an elementary matrix of the same type.