1.4 THE MATRIX EQUATION Ax = b

The equation Ax=b is consistent if the augmented matrix [ A b ] has a pivot position in every row. T/F?

False. If the augmented matrix [ A b ] has a pivot position in every​ row, the equation equation Ax=b may or may not be consistent. One pivot position may be in the column representing b.

The equation Ax=b is referred to as a vector equation. T/F?

False. The equation Ax=b is referred to as a matrix equation because A is a matrix.

If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in ℝ^m​, then A cannot have a pivot position in every row. T/F?

True. If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in ℝ^m​, then the equation Ax=b has no solution for some b in ℝ^m.

If the columns of an mxn matrix A span ℝ^m​, then the equation Ax=b is consistent for each b in ℝ^m. T/F?

True. If the columns of A span ℝ^m, then the equation Ax=b has a solution for each b in ℝ^m.

A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution. T/F?

True. The equation Ax=b has the same solution set as the equation x(1) a(1) + x(2) a(2) + ... + x(n) a(n) = b.

The first entry in the product Ax is a sum of products. T/F?

True. The first entry in Ax is the sum of the products of corresponding entries in x and the first entry in each column of A.