Question 23: Mark each statement True or False. Justify each answer.

a. The equation\(Ax = b\)is referred to as a vector equation.

b. A vector\(b\)is a linear combination of the columns of a matrix A if and only if the equation\(Ax = b\)has at leastone solution.

c. The equation\(Ax = b\)is consistent if the augmented matrix\(\left[ {\begin{array}{*{20}{c}}A&b\end{array}} \right]\)has a pivot position in every row.

d. The first entry in the product\(Ax\)is a sum of products.

e. If the columns of an\(m \times n\)matrix A span\({R^m}\), then the equation\(Ax = b\)is inconsistent for some b in\({R^m}\).

f. If A is an \(m \times n\) matrix and if the equation \(Ax = b\)is inconsistent for some b in \({R^m}\), then A cannot have a pivotposition in every row.

The equation \(Ax = b\) is known as the matrix equation.

Hence, the given statement is false.

According to the existence of the solutions, If and only if b is a linear combination of A's column, then the equation \(Ax = b\) has a solution.

Hence, the given statement is true.

The equation \(Ax = b\) may or may not be consistent if an augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&b\end{array}} \right]\) has a pivot position in every row.

Hence, the given statement is false.

The row vector rule states that the sum of the products of comparable elements from row \(i\) of A and from vector x equals the \(ith\) entry in \(Ax\) if the product \(Ax\) is specified.

Hence, the given statement is true.

Assume A is a \(m \times n\) matrix.Then,

a. The equation \(Ax = b\) has a solution for each b in \({R^m}\).

b. \({R^m}\) is spanned by the columns of A.

Hence, the given statement is true.

Assume A is a \(m \times n\) matrix. Thus, the propositions that follow are logically equivalent if:

a. The equation \(Ax = b\) has a solution for each b in \({R^m}\).

b. In each row, A has a pivot position.

Hence, the given statement is true.