What Does Triangularizing a Matrix Mean in Linear Algebra?

[kolycholy]

what does triangularizing a matrix mean?
i am supposed to find determinant of a 3 by 3 matrix using big formula, but then I am asked to verify my result by "triangularizing the matrix"
thanks!

 

[HallsofIvy]

Use row operations to reduce the matrix to "upper triangular". That is, a matrix that has only 0's below the main diagonal. You do not need (nor want) to get 1 on the diagonal. If you reduce a matrix to upper triangular using only the row operations of "swap two rows" and "add (or subtract) a multiple of one row to (from) another", and not "multiply (or divide) one row by a number", then the determinant is just the product of the numbers on the main diagonal.

 

[tacman]

Triangularizing a matrix means transforming it into a triangular form, where all the entries below the main diagonal are zero. This is typically done by using elementary row operations such as row swapping, scaling and addition.

In the context of finding the determinant of a 3x3 matrix, triangularizing the matrix can help to simplify the calculation of the determinant. This is because the determinant of a triangular matrix is simply the product of the entries along the main diagonal.

In your case, you can use the "big formula" to calculate the determinant of the 3x3 matrix, but verifying your result by triangularizing the matrix is a good way to double check your answer. If the determinant obtained from the triangular form matches the one calculated using the formula, then you can have more confidence in your solution.

In summary, triangularizing a matrix is a useful technique in linear algebra that can help simplify calculations and verify results. I hope this helps clarify the concept for you.