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Laplace Expansion Equation & Finding Determinants

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  • 0:03 Laplace Expansion Equation
  • 0:34 Breaking Down the Equation
  • 2:51 Using the LEE When N = 2
  • 4:39 Using the LEE When N = 3
  • 5:22 Mathematical Operations
  • 7:51 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

It is fascinating how a single number can tells us something about a matrix. One such number is the determinant. In this lesson we find determinants using the Laplace expansion equation.

Laplace Expansion Equation

Have you played checkers or chess? The alternating colors on a checkerboard are reminiscent of the alternating signs in the Laplace expansion equation for finding determinants.


A game of checkers.
Picture of a checker board.


In this lesson, we explore the Laplace expansion equation (LEE), a method that uses determinants of smaller matrices to find the determinant of a larger square matrix. We can use the Laplace expansion equation to find determinants of progressively larger matrices, and, like a checkerboard strategy, we can test this method's efficiency.

Breaking Down the Equation

The Laplace expansion equation is written as:


LEE.


What does all that mean? Let's start at the left with the determinant of A. The determinant is a number representing the matrix A. Determinants are used for matrix inverses, Jacobians, and geometry. The matrix A is a square matrix, an equal number of rows and columns, sometimes called an n-by-n matrix (written as 'nxn').

'is equal to the sum over j' : We let j be 1 all the way to n. For each j, we evaluate the expression to the right of the Σ. Then we add each of those results together. The order of the square matrix is n, which is the number of rows (or columns) in A.

'of ai,j': The a is one of the numbers in the matrix A. The i and j give the row number and the column number location for a. The matrix is like a checkerboard where the top-left square is the (i = 1, j = 1) location.

'times -1 to the i plus j': -1 is raised to the power of i plus j. When the power is an even number, we get +1. When the power is an odd number, we get -1. The signs alternate like the colors on the checkerboard.

'times Mi,j': The minor, M is the determinant of a smaller matrix within the matrix A. This will become clearer as we do examples.

'for i, 1 to n': The row number is the value we select for n. No matter which row we choose from 1 up to n, we get the same final result for the Laplace expansion equation.

If there is a row or column with zeros in it, selecting that row or column reduces the number of computations because multiplying by a zero is zero; one less minor to compute. Thus, we want to select a column instead of a row. This simply means switching all the i and the j in the Laplace expansion equation.

You may see the Laplace expansion equation expressed with a C instead of the M. This C is the cofactor, and it's the minor M with the (-1)^(i+j).

Time to return to the checkerboard.

Using the LEE When N = 2

We begin small and gradually get bigger.

What if we had a 2x2 matrix (order n = 2) like this:


order2.


You might know how to find the determinant. Multiplying along diagonals gives us 4(6) - 1(3) = 21.


order2 det.


The Laplace expansion equation is essentially the same, but it's more formal. Let's put in the details. We choose one of the two rows. For instance, we choose the first row: i = 1. Then, with the i = 1 and n = 2, we expand the equation:


expandLEE.


In the second line, we plug in n = 2 for our two columns. In the third line, we have a term for j = 1, and another for j = 2. In the fourth, we substitute 4 and 1 for the matrix elements at row 1, column 1 and row 2, column 2. And in the fifth line, we simplify the exponents on the -1 terms.

Vertical bars are another way to write 'det( )'. To find the minors, imagine locating row i = 1 and column j = 1. This is the top-left corner of the matrix. The number 4 is there. Just like a rook (a castle) in the game of chess, you move horizontally along the rows and vertically along the columns. Imagine deleting all the numbers in the row and column identified by the location (1,1).


M11.


This leaves a smaller matrix with only the number 6 in it. We take the determinant and get the minor. The determinant of a matrix with one number is just the number. The M1,1 minor is 6. How about M1,2? Locate the (1,2) position, which is row 1 and column 1, the top-right corner. Delete the row and column, leaving a matrix whose determinant is 3.


M12.


Thus:


LEEfinish.


Are you ready for a larger checkerboard?

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