4.2: Laplace Expansion and Leibniz Formula

View Laplace Expansion for Computing Determinants on YouTube

View Leibniz Formula for Computing Determinants on YouTube

There are two ways to view the three-by-three determinant that do in fact generalize to \(n\)-by-\(n\) matrices. The first way writes

\[\begin{aligned}\det\text{A}&=aei+bfg+cdh-ceg-bdi-afh \\ &= a(ei − f h) − b(di − f g) + c(dh − eg) \\ &=a\left|\begin{array}{cc}e&f\\h&i\end{array}\right| -b\left|\begin{array}{cc}d&f\\g&i\end{array}\right|+c\left|\begin{array}{cc}d&e\\g&h\end{array}\right|.\end{aligned} \nonumber \]

The three-by-three determinant is found from lower-order two-by-two determinants, and a recursive definition of the determinant is possible. This method of computing a determinant is called a Laplace expansion, or cofactor expansion, or expansion by minors. The minors refer to the lower-order determinants, and the cofactor refers to the combination of the minor with the appropriate plus or minus sign. The rule here is that one goes across the first row of the matrix, multiplying each element in the first row by the determinant of the matrix obtained by crossing out the element’s row and column. The sign of the terms alternate as we go across the row.

Instead of going across the first row, we could have gone done the first column using the same method to obtain

\[\det\text{A}=a\left|\begin{array}{cc}e&f \\ h&i\end{array}\right|-d\left|\begin{array}{cv}b&c\\h&i\end{array}\right|+g\left|\begin{array}{cc}b&c\\e&f\end{array}\right|,\nonumber \]

also equivalent to (4.1.1). In fact, this expansion by minors can be done across any row or down any column. The sign of each term in the expansion is given by \((−1)^{i+j}\) when the number multiplying each minor is drawn from the \(i\)th-row and \(j\)-th column. An easy way to remember the signs is to form a checkerboard pattern, exhibited here for the three-by-three matrix:

\[\left(\begin{array}{ccc}+&-&+\\-&+&-\\+&-&+\end{array}\right).\nonumber \]

The second way to generalize the determinant is called the Leibniz formula, or more descriptively, the big formula. One notices that each term in (4.1.1) has only a single element from each row and from each column. As we can choose one of three elements from the first row, then one of two elements from the second row, and only one element from the third row, there are \(3! = 6\) terms in the expansion. For a general \(n\)-by-\(n\) matrix there are \(n!\) terms.

The sign of each term depends on whether it derives from an even or odd permutation of the columns numbered \(\{1,\: 2,\: 3,\cdots , n\}\), with even permutations getting a plus sign and odd permutations getting a minus sign. An even permutation is one that can be obtained by switching pairs of numbers in the sequence \(\{1,\: 2,\: 3,\cdots , n\}\) an even number times, and an odd permutation corresponds to an odd number of switches. As examples from the three-by-three case, the terms \(aei,\: bfg,\) and \(cdh\) correspond to the column numberings \(\{1,\: 2,\: 3\},\: \{2,\: 3,\: 1\}\), and \(\{3,\: 1,\: 2\}\), which can be seen to be even permutations of \(\{1,\: 2,\: 3\}\), and the terms \(ceg,\: bdi,\) and \(afh\) correspond to the column numberings \(\{3,\: 2,\: 1\},\: \{2,\: 1,\: 3\},\) and \(\{1,\: 3,\: 2\}\), which are odd permutations.

Either the Laplace expansion or the Leibniz formula can be used to define the determinant of an \(n\)-by-\(n\) matrix.

It will, however, be more fun to define the determinant from three more fundamental properties. These properties will lead us to the most important condition \(\det\text{A}\neq 0\) for an invertible matrix. But we will also elucidate many other useful properties.