Discover Linear Algebra: A first course in linear algebra

It may seem from Section 8.2 that the definition of determinant is circular — we define the determinant in terms of entries and cofactors (via cofactor expansions), where cofactors are defined in terms of minors, which are defined in terms of … determinants? But the key word in the definition of minor is smaller — determinants are defined recursively in terms of smaller matrices. In Discovery guide 8.1, after first exploring the determinant of a $2\times 2$ matrix as motivation, we started afresh with a precise definition of the $1\times 1$ determinant, and then defined the $2\times 2$ determinant in terms of $1\times 1$ determinants. Then the $3\times 3$ determinant is defined in terms of $2\times 2$ determinants, and so on. As we will see in examples in Section 8.4, computing a determinant from this recursive definition will involve unpacking it in terms of determinants of one smaller size, then unpacking those in terms of determinants of one size smaller again, and so on. Technically, this process should continue until we are down to a bunch of $1\times 1$ determinants, but since there is a simple formula for a $2\times 2$ determinant, in direct computations we will stop there.

Consider the general $1\times 1$ matrix $A=\left[\begin{array}{c}a\end{array}\right]\text{.}$ We should expect the invertibility of $A$ to be completely determined by the value of the single entry $a\text{,}$ since that is all the information that $A$ contains. And that is precisely the case, as $A$ is invertible when $a\ne 0\text{,}$ with $A^{-1}=\left[\begin{array}{c}{a}^{-1}\end{array}\right]\text{,}$ and $A$ is singular when $a=0\text{,}$ because then $A$ would be the zero matrix. Since entry $a$ determines the invertibility of $A\text{,}$ we set $det\left[\begin{array}{c}a\end{array}\right]=a\text{.}$

An easy way to remember the $2\times 2$ determinant formula is with a crisscross pattern, as illustrated in Figure 8.3.4 for general $2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{.}$

In Figure 8.3.4, the plus and minus signs do not mean positive and negative, but instead mean add and subtract: we start with a value of $0\text{,}$ then add the product of the two entries along the arrow labelled $+$ (which may be a negative number) and subtract the product of the two entries along the arrow labelled $-\text{.}$

While we have a convenient general formula for $2\times 2$ matrices in terms of the four entry variables, we certainly wouldn’t want to attempt to write out a general formula for the determinant of a $5\times 5$ matrix in twenty-five entry variables. Instead, for matrices larger than $2\times 2\text{,}$ computing a determinant for a specific matrix from a cofactor expansion is a recursive process, since cofactors are just minor determinants with some sign changes. A cofactor expansion for an $n\times n$ matrix requires $n$ cofactor calculations. Each of those cofactor calculations is a determinant calculation of some $(n-1)\times (n-1)$ “submatrices”. Each of those determinants, if calculated by cofactor expansion, will require $n-1$ determinant calculations of various $(n-2)\times (n-2)$ “submatrices”. And so on. As you can see, the number of calculations involved grows out of hand quite quickly, even for single-digit values of $n\text{.}$

When computing the determinant of an upper triangular matrix, a similar pattern of computation as in the diagonal case would arise, because choosing to always expand along the first column would result in diagonal entry times an upper triangular minor determinant. And the same pattern would repeat for lower triangular matrices, but for those it is best to expand along the first row.