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Matrix transformations involve altering a matrix's structure or elements through specific operations. These operations can include scaling the matrix's elements by a constant, swapping rows or columns, and even multiplying by another matrix. When you change the elements of a matrix, you essentially apply a transformation.

In our exercise, multiple transformations were applied to the original matrix. Each item in the first column was multiplied by 2, effectively scaling it. The second column's elements were divided by 3, reducing their magnitude, and this is a form of linear scaling too. Lastly, the third column had its elements multiplied by -1, which switches their signs.

• Scaling: Multiplying or dividing a whole row or column by a number.
• Negating: Changing the sign of elements in a row or column.
• Swapping: Exchanging positions of rows or columns — not used in this exercise, but useful to note.

Understanding these basic transformations can help in determining how they affect the structure and properties of the matrix, including the determinant.

The determinant is a special number associated with a square matrix. It provides insights into the matrix's properties such as whether the matrix is invertible and its scaling factor for volume transformations.

Several key properties pertain to how determinants react to transformations:

• If you multiply a row or column by a constant, the determinant is also multiplied by that constant.
• If you switch two rows, the determinant's sign changes (it is multiplied by -1).
• The determinant is zero if two rows or columns are identical or if a row or column is all zeros.

In our exercise, each column's transformation in the matrix is a perfect example of these properties:

• The first column's elements are all scaled by 2, thus multiplying the determinant by 2.
• The second column's elements are divided by 3, influencing the determinant by a factor of \( \frac{1}{3} \).
• The third column's elements are multiplied by -1, flipping the sign of the determinant.

These rules highlight how powerful determinant properties are in simplifying calculations and understanding matrix behavior.

Linear algebra deals with calculating aspects of matrices and linear transformations. Such calculations are fundamental to numerous fields, including physics, engineering, computer science, and economics.

When working with determinants in linear algebra, calculations often require understanding how transformations affect the matrix. Here's a breakdown of what we calculate in our particular problem:

• Step 1: Knowing the original determinant value helps start the calculation.
• Step 2: Understanding the effects of the scaling factors: each column was altered by multiplying it by a constant. The total effect is the product of all these changes.
• Step 3: Apply these combined effects to the original determinant value.

Therefore, following these modifications, we arrive at the overall determinant of the new matrix, which is \( \left(-\frac{2}{3} \times 4 = -\frac{8}{3}\right) \).

This demonstrates the sequential thought process of dealing with matrix transformations using linear algebra. It showcases how methodical steps lead to precise results.