Choose which of the following are true statements regarding matrices.
a. A matrix with 14 elements is impossible.
b. If two rows are identical in a matrix, then there are infinitely many solutions.
c. A 3-by-2 matrix and a 2-by-3 matrix have the same number of elements.
d. An acceptable row operation is to switch columns.
The true statements regarding matrices are:
b. If two rows are identical in a matrix, then there are infinitely many solutions.
c. A 3-by-2 matrix and a 2-by-3 matrix have the same number of elements.
To determine which of the statements are true regarding matrices, let's analyze each statement one by one:
a. A matrix with 14 elements is impossible.
To determine if this statement is true or false, we need to consider the number of elements in a matrix. The number of elements in a matrix is equivalent to the product of its number of rows and the number of columns. Therefore, the possible numbers of elements in a matrix are 0, 1, 2, 3, 4, 5, 6, 7, 8, etc. There is no restriction on the number of elements, so a matrix with 14 elements is possible. Therefore, statement a is false.
b. If two rows are identical in a matrix, then there are infinitely many solutions.
This statement is related to systems of linear equations represented by matrices. When two rows are identical in a matrix, it means that the corresponding linear equations are dependent, and there are infinitely many solutions. This is due to the fact that one equation is redundant and doesn't provide any new information. Therefore, statement b is true.
c. A 3-by-2 matrix and a 2-by-3 matrix have the same number of elements.
To determine if this statement is true or false, we need to calculate the number of elements in each matrix. The number of elements in a matrix is equal to the product of its number of rows and the number of columns. In the case of a 3-by-2 matrix, we have 3 rows and 2 columns, giving us a total of 3 * 2 = 6 elements. For a 2-by-3 matrix, we have 2 rows and 3 columns, also resulting in 2 * 3 = 6 elements. Therefore, statement c is true.
d. An acceptable row operation is to switch columns.
When performing row operations on a matrix, it is crucial to focus on manipulating the rows rather than the columns. Row operations involve multiplying a row by a nonzero number, adding or subtracting rows, or switching rows. However, switching columns is not considered an acceptable row operation. Therefore, statement d is false.
Overall, the true statements regarding matrices are statement b and statement c, while statements a and d are false.
a. A matrix with 14 elements is possible.
b. If two rows are identical in a matrix, then there are infinitely many solutions. (This statement is not necessarily true. It depends on the specific system of equations represented by the matrix.)
c. A 3-by-2 matrix and a 2-by-3 matrix do have the same number of elements.
d. An acceptable row operation is to switch columns. (This statement is not true. An acceptable row operation involves only the manipulation of rows, not columns.)