Value of determinant is computed by adding multiples of one row to another row

To understand why the value of a determinant is computed by adding multiples of one row to another row, let's first review what a determinant is and how it is calculated.

A determinant is a mathematical value that can be calculated for a square matrix. It provides important information about the matrix, such as whether it is invertible or singular. The determinant of a matrix A is denoted as |A| or det(A).

The determinant of a 2x2 matrix is calculated using the formula:
|A| = a*d - b*c

For a 3x3 matrix, the determinant is calculated using the formula:
|A| = a*(e*i - f*h) - b*(d*i - f*g) + c*(d*h - e*g)

Similarly, for larger matrices, the determinant is calculated using more complex formulas involving the elements of the matrix.

Adding multiples of one row to another row

When computing the determinant, we can use row operations to simplify the calculation. One common row operation is adding multiples of one row to another row. This operation does not change the value of the determinant.

Explanation

When we add multiples of one row to another row, we are essentially performing a linear combination of the rows. This means that we are multiplying each element in one row by a constant and then adding the corresponding elements in another row.

By performing this operation, we are not changing the linear dependence between the rows. The determinant measures the linear dependence of the rows (or columns) of a matrix. Therefore, adding multiples of one row to another row does not affect the determinant.

This property allows us to simplify the calculation of the determinant by using row operations to transform the matrix into an upper triangular form or lower triangular form. In these forms, the determinant can be easily calculated by multiplying the elements on the main diagonal.

Conclusion

In conclusion, the value of a determinant is computed by adding multiples of one row to another row. This is possible because adding multiples of one row to another row does not change the linear dependence between the rows of a matrix. By using row operations, we can simplify the calculation of the determinant and obtain its value.