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When working with matrices, it's essential to comprehend the significance of matrix dimensions. The dimensions of a matrix tell us the number of rows and columns it contains, often written as 'rows x columns'. For instance, if we talk about a matrix with the dimensions of (2 x 3), like the matrices B and C in the example, it means the matrix has 2 rows and 3 columns.

Checking the dimensions is the first step in any matrix operation because it determines whether the operation can be performed. For subtraction and addition, the involved matrices must have identical dimensions. In our exercise, both B and C are (2 x 3) matrices, so they can be subtracted element-wise. If the dimensions do not match, the subtraction simply cannot be done.

The field of matrix arithmetic involves operations such as addition, subtraction, and multiplication on matrices. While these operations are analogous to arithmetic with numbers, they follow specific rules due to the nature of matrices. Let's briefly look at those rules:

• Addition or Subtraction: These operations are possible only if the matrices involved are of the same size, and you perform these operations element by element.
• Multiplication by a scalar: You can multiply every element of a matrix by a number (scalar).
• Matrix Multiplication: To multiply two matrices, the number of columns in the first must equal the number of rows in the second.

Understanding these basics is crucial before you proceed with any complex matrix operations.

When it comes to subtracting matrices, the procedure is straightforward if we are adhering to the rules of matrix arithmetic. Let's break down the process:

First, ensure the matrices share the same dimensions. Once verified, you subtract elements in the same position from each other. The resulting matrix will have the same dimensions as the original matrices.

• To subtract matrix C from matrix B, you would take each element in B and subtract the corresponding element in C.
• The operation is carried out element-wise, meaning you compute the difference for each pair of elements separately to form the new matrix.

In our example, we found the resulting matrix by subtracting each element of C from the corresponding element of B, which is only possible because their dimensions match.