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When you're diving into matrix multiplication, understanding matrix dimensions is your starting point. For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This requirement is essential as it determines whether the multiplication is possible or not.

For example, consider matrix \(C\) with dimensions \(2 \times 3\) and matrix \(B\) with dimensions \(3 \times 2\). The number of columns in \(C\) (which is 3) perfectly matches the number of rows in \(B\) (also 3), making the product \(CB\) feasible. If these dimensions do not align, the matrices cannot be multiplied. This rule is crucial when dealing with matrix operations in linear algebra.

In summary, always check:

• The number of columns in the first matrix.
• The number of rows in the second matrix.
• If they are equal, multiplication is possible!

Once you've confirmed that matrix multiplication is possible, the next step is to determine the dimensions of the resulting matrix. This resulting matrix will have a distinct set of dimensions derived from the original matrices.

The dimensions of the new matrix are determined as follows:

• The number of rows in the resulting matrix equals the number of rows of the first matrix.
• The number of columns equals the number of columns of the second matrix.

For the matrices \(C\) and \(B\), which are \(2 \times 3\) and \(3 \times 2\) respectively, the resulting matrix \(CB\) will thus be \(2 \times 2\). This is a direct result of having two rows from \(C\) and two columns from \(B\).

Understanding the resulting matrix dimensions is crucial as it sets up the structure for filling in the elements during matrix multiplication.

Matrix operations, especially multiplication, involve a methodical approach to filling elements in the resulting matrix. With the previous steps confirming the feasibility and dimensions of the product \(CB\), you're ready to perform the actual operations.

Here's how multiplication works in this context:1. Multiply rows by columns:
- For each element in the resulting matrix, take a row from the first matrix and a column from the second matrix.
- Perform a dot product; that is, multiply corresponding elements and sum them up.
2. Fill each position in the resulting matrix:
- For example, to find the top-left element of \(CB\), multiply the first row of \(C\) by the first column of \(B\): \((-5 \times 5) + (4 \times 0) + (1 \times 3) = -25 + 0 + 3 = -22\).
- Continue this pattern for each position in the resulting matrix.

This systematic approach helps in achieving accurate calculations, ensuring each element in the resulting matrix is computed correctly. These operations are at the heart of working with matrices and are widely used in various mathematical and real-world applications.