If and are matrices, and is a matrix, which of the following are defined?

General
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if and are matrices, and is a matrix, which of the following are defined?

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If and are matrices, and is a matrix, it is important to understand what is meant by “defined”. In the context of matrices, “defined” typically refers to the operations that can be performed with the given matrices.

Here are several common operations involving matrices:

  1. Matrix Addition: The addition of two matrices is defined if they have the same dimensions. In other words, if and have the same number of rows and the same number of columns, then the sum of and, denoted as , is a defined operation.

    • Example: If and have dimensions , then their sum, , will also have dimensions .

  2. Matrix Subtraction: Similar to addition, the subtraction of two matrices is defined if they have the same dimensions. In other words, if and have the same number of rows and the same number of columns, then the difference of and, denoted as , is a defined operation.

    • Example: If and have dimensions , then their difference, , will also have dimensions .

  3. Scalar Multiplication: Scalar multiplication involves multiplying a matrix by a scalar (a single number). It is defined for any matrix and any scalar value .

    • Example: If is a matrix and is a scalar, then the scalar multiplication of and, denoted as , is a defined operation.

  4. Matrix Multiplication: Matrix multiplication is defined if the number of columns in matches the number of rows in . In other words, if has dimensions and has dimensions , then the product of and, denoted as , is a defined operation.

    • Example: If has dimensions and has dimensions , then their product, , will have dimensions .

  5. Transpose: The transpose of a matrix is defined for any matrix . It involves swapping the rows and columns of the original matrix, resulting in a new matrix denoted as .

    • Example: If is a matrix, then the transpose of , denoted as , is a defined operation.

It is important to note that not all operations are defined for all combinations of matrices. For example, matrix addition requires the matrices to have the same dimensions, while matrix multiplication has specific requirements for the number of rows and columns. It is necessary to consider the dimensions and properties of the matrices involved to determine if a specific operation is defined.