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Section 4.1 The determinant of small matrices

Each square matrix of order \(n\) has a number associated with it called its determinant. A matrix \(A\) has its determinant denoted by \(\det(A).\) For small values of \(n\text{,}\) we can give a straightforward method of evaluation.

Definition 4.1.1. Determinants of small matrices.

Let \(A\) be a square matrix of order \(n\text{.}\) Then

  • If \(n=1\text{,}\) and \(A=[a_{1,1}]\) then \(\det(A)=a_{1,1}.\)

  • If \(n=2\) and \(A=\begin{bmatrix}a_{1,1}\amp a_{1,2}\\a_{2,1}\amp a_{2,2}\end{bmatrix}\text{,}\) then \(\det(A)=a_{1,1}a_{2,2}-a_{1,2}a_{2,1}.\)

  • If \(n=3\) and \(A=\begin{bmatrix}a_{1,1}\amp a_{1,2}\amp a_{1,3}\\ a_{2,1}\amp a_{2,2}\amp a_{2,3}\\ a_{3,1}\amp a_{3,2}\amp a_{3,3}\end{bmatrix}\text{,}\) then \(\det(A)=a_{1,1}a_{2,2}a_{3,3}+a_{1,2}a_{2,3}a_{3,1} +a_{1,3}a_{2,1}a_{3,2} -a_{1,1}a_{2,3}a_{3,2} -a_{1,2}a_{2,1}a_{3,3} -a_{1,3}a_{2,2}a_{3,1} \text{.}\)

Example 4.1.2. Determinants of small matrices.

  • \(\displaystyle \det\begin{bmatrix}1\amp2\\3\amp4\end{bmatrix}=4-6=-2.\)

  • \(\displaystyle \det \begin{bmatrix} 1\amp2\amp3\\ 3\amp4\amp-1\\ 1\amp2\amp1 \end{bmatrix} =4-2+18-(-2)-6-12=4.\)

There is a nice way to visualize these determinants by considering the diagonals:

For \(n=2\text{,}\) \(A=\begin{bmatrix}\color{red} {a_{1,1}}\amp\color{green}{a_{1,2}}\\ \color{green}{a_{2,1}}\amp\color{red}{a_{2,2}}\end{bmatrix}\) and the determinant is the product of the two diagonal elements minus the product of the other two elements. \(\det(A)= {\color{red}{a_{1,1}}\color{red}{{a_{2,2}}}-\color{green}{a_{1,2}a_{2,1}}}.\)

For \(n=3\text{,}\) we may take \(A\) and append a copy of the first two columns to the right side. Then the determinant is the sum of six terms: the three positive ones are the product of terms on the upper-left to lower-right diagonals while the three negative ones are the product of terms on the lower-left to upper-right diagonals.

\begin{equation*} \begin{array}{c c} \begin{bmatrix} \color{red}{a_{1,1}}\amp\color{green}{a_{1,2}}\amp\color{blue}{a_{1,3}}\amp a_{1,1}\amp a_{1,2}\\ a_{2,1}\amp\color{red}{a_{2,2}}\amp\color{green}{a_{2,3}}\amp\color{blue}{a_{2,1}}\amp a_{2,2}\\ a_{3,1}\amp a_{3,2}\amp\color{red}{a_{3,3}}\amp\color{green}{a_{3,1}}\amp\color{blue}{a_{3,2}} \end{bmatrix} \amp \begin{bmatrix} a_{1,1}\amp a_{1,2}\amp\color{red}{a_{1,3}}\amp\color{green}{a_{1,1}}\amp\color{blue}{a_{1,2}}\\ a_{2,1}\amp\color{red}{a_{2,2}}\amp\color{green}{a_{2,3}}\amp\color{blue}{a_{2,1}}\amp a_{2,2}\\ \color{red}{a_{3,1}}\amp\color{green}{a_{3,2}}\amp\color{blue}{a_{3,3}}\amp a_{3,1}\amp a_{3,2} \end{bmatrix}\\ \text{(positive terms are in same colour)} \amp \text{(negative terms are in same colour)} \end{array} \end{equation*}

Alternatively, by letting the diagonals wrap around to from the right side of the matrix to the left side for the first matrix, and from the left side to the right side in the second matrix.

\begin{equation*} \begin{array}{c c} \begin{bmatrix} \color{red}{a_{1,1}}\amp\color{green}{a_{1,2}}\amp\color{blue}{a_{1,3}}\\ \color{blue}{a_{2,1}}\amp\color{red}{a_{2,2}}\amp\color{green}{a_{2,3}}\\ \color{green}{a_{3,1}}\amp \color{blue}{a_{3,2}}\amp\color{red}{a_{3,3}} \end{bmatrix} \amp \begin{bmatrix} \color{green}{a_{1,1}}\amp\color{blue}{a_{1,2}}\amp\color{red}{a_{1,3}}\\ \color{blue}{a_{2,1}}\amp\color{red}{a_{2,2}}\amp\color{green}{a_{2,3}}\\ \color{red}{a_{3,1}}\amp\color{green}{a_{3,2}}\amp\color{blue}{a_{3,3}} \end{bmatrix}\\ \text{(positive terms are in same colour)} \amp\text{(negative terms are in same colour)} \end{array} \end{equation*}

However visualized, we can evaluate \(\det(A)= ({\color{red}{a_{1,1}a_{2,2}a_{3,3}}} +{\color{green}{a_{1,2}a_{2,3}a_{3,1}}} +{\color{blue}{a_{1,3}a_{2,1}a_{3,2}}}) -({\color{green}{a_{1,1}a_{2,3}a_{3,2}}} +{\color{blue}{ a_{1,2}a_{2,1}a_{3,3}}} +{\color{red}{a_{1,3}a_{2,2}a_{3,1}}}).\) So it's pretty easy to compute the determinant of a matrix of order \(n\) when \(n\leq 3\text{.}\) Unfortunately, there is no straightforward generalization for \(n\gt3\text{.}\)

There is a theoretical formula for the evaluation of the determinant for larger \(n\text{,}\) but the number of summands grows quickly (for \(n=4\) there are 24 terms to add, and for \(n=5\) there are 120). The present goal is to find better methods to evaluate the determinant. The first technique involves objects called cofactors, and so we need to make a digression to define and to see how to use them.