If A and B are 4 * 6 matrices and C is a 7 * 4 matrix, which of the following are defined? 1. CA.2. B^T * C^T.3. AC.4. A-B.5. B-C.6. B^T.

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

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