Respuesta :
Answer:
1. CA is defined
2. B^T * C^T defined
3. AC is NOT defined
4. A - B is defined
5. B - C is NOT defined
6. B^T is defined
Hence, 1,2,4, and 6 are defined
Step-by-step explanation:
Order of the matrices:
Matrix A is a (4 × 6) matrix
Matrix B is a (4 × 6) matrix
Matrix C is a (7 × 4) matrix
To determine which are defined
1. CA
CA means product of Matrix C and Matrix A
To determine the product of two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. For example, given that matrix X is a (m × n) matrix, matrix Y is a (n × k) matrix and matrix Z is (k × n) matrix. From the definition above,
XY can be determined because number of columns in X equals number of rows in Y
XZ cannot be determine because number of columns in X is not equal to the number of rows in Z.
YZ can be determined because number of columns in Y equals number of rows in Z
(NOTE: A matrix of order (m × n) means the matrix has m rows and n columns)
Hence, for CA
Matrix C is a (7 × 4) matrix
Matrix A is a (4 × 6) matrix
Number of columns in C equals number of rows in A, hence CA is defined
2. B^T * C^T
B^T means transpose of matrix B
Transpose of a matrix is defined as a new matrix whose rows are the same as the columns of the original matrix and whose columns are the rows of the original matrix. Hence, for a matrix of order (m × k), the transpose of the matrix will have an order (k × m).
Hence,
Matrix B is a (4 × 6) matrix
∴B^T gives a matrix of order (6 × 4)
Matrix C is a (7 × 4) matrix
∴ C^T gives a matrix of order (4 × 7)
Since the number of columns in B^T equals the number of rows in C^T, then B^T * C^T defined
3. AC
Matrix A is a (4 × 6) matrix
Matrix C is a (7 × 4) matrix
The number of columns in A is NOT equal to the number of rows in C, hence AC is NOT defined
4. A - B
To find the difference between two matrices, the matrices must have the same order.
Matrix A is a (4 × 6) matrix
Matrix B is a (4 × 6) matrix
Since matrices A and B have the same order [(4 × 6)], hence A - B is defined
5. B - C
Matrix B is a (4 × 6) matrix
Matrix C is a (7 × 4) matrix
Since matrices B and C are of different orders [(4 × 6) and (7 × 4) respectively], then B - C is NOT defined
6. B^T
B^T means the transpose of matrix B
Matrix B is a (4 × 6) matrix
From the above definition
B^T gives a (6 × 4) matrix
Hence, B^T is defined
