What is the Most Efficient Method for Finding the Determinant of an nxn Matrix?

In summary, the student is struggling with a problem involving finding the determinant using Laplace expansion. The professor advised against using this method and suggested adding rows and using indices instead. The student is not familiar with this approach and is finding the textbook unhelpful.
  • #1
JoshW
9
0

Homework Statement


Shown In the picture. I went to the prof for help he said and i quote :" don't use laplas expansion to find the determinate, it will take you for ever."

Homework Equations


I don't even know how to do this. prof had no notes on this and Boas is a god awful book for learning anything.

The Attempt at a Solution


Add every row to the last row. 1/last row - every other row. then every row minus the last. but my prof wanted it solved with indices. I don't know what that is or how to apply it as it has never came up.
 

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  • #2
Adding one row to another and eliminate the order of the matrix, which may help.
 

What is the determinant of a nxn matrix?

The determinant of a nxn matrix is a numerical value that can be computed from the elements of the matrix. It is a fundamental property of a square matrix and is often used in various mathematical and scientific applications.

How is the determinant of a nxn matrix calculated?

The determinant of a nxn matrix is calculated by using a specific formula that involves finding the products of certain elements in the matrix. The exact formula can vary depending on the size of the matrix, but it typically involves subtracting the products of diagonally opposite elements in the matrix.

What does the determinant of a nxn matrix represent?

The determinant of a nxn matrix represents several important properties of the matrix. These include the volume of a parallelepiped formed by the column vectors of the matrix, the number of linearly independent rows or columns in the matrix, and the factor by which the matrix scales the area of a unit square.

Why is the determinant of a nxn matrix important?

The determinant of a nxn matrix is important because it helps determine if a matrix has an inverse and if a system of linear equations has a unique solution. It is also used in various mathematical and scientific fields, such as in solving differential equations, calculating eigenvalues, and in geometry.

Can the determinant of a nxn matrix be negative?

Yes, the determinant of a nxn matrix can be negative. It can take on any real value, including positive, negative, or zero. The sign of the determinant can provide information about the orientation of the vectors in the matrix, but it does not affect the magnitude of the determinant itself.

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