Chapter 7: Determinants

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The determinant is a function from a matrix $\mx{A}$ to a value $\det(\mx{A})$. We will later see that the determinant can be defined using three simple properties. However, before delving into the details we will first look at explicit expressions for the determinant in dimensions one, two and three. For $1 \times 1$ matrices

\begin{equation} \mx{A} = \left( \begin{array}{r} a \end{array} \right) \end{equation}

(7.3)

the determinant is simply $\det(\mx{A}) = a$. This is the length, with sign, of the vector

\begin{equation} \vc{a} = \left( \begin{array}{r} a \end{array} \right) . \end{equation}

(7.4)

Here it is straightforward to see that the equation $ \mx{A} \vc{x} = \vc{b} $ has a unique solution when $\det(\mx{A}) = a \neq 0$. When the determinant is zero, there could be no solutions, if $\vc{b} \neq \vc{0}$. As an example for $1\times 1$ matrices, i.e., where the equation reduces to $ax=b$, assume that $a x = 3$ has no solutions if $a=0$. If $b = 0 $, there are infinitely many solutions $x$ since the equation now reads $0 x = 0$.

For a $2 \times 2$ matrix

\begin{equation} \mx{A} = \left( \begin{array}{rr} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array} \right) \end{equation}

(7.5)

the determinant is $\det(\mx{A}) = a_{11} a_{22} - a_{12} a_{21}$. The determinant is sometimes written with a single straight line as left and right brackets, thus we would write the above as

\begin{equation} \det(\mx{A}) = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11} a_{22} - a_{12} a_{21} . \end{equation}

(7.6)

If we think of the two columns as the coordinates of two vectors in the plane, i.e.,

\begin{equation} \mx{A} = \begin{pmatrix} \vc{a}_{,1} & \vc{a}_{,2} \end{pmatrix} \end{equation}

(7.7)

then the determinant is the area (with sign) of the parallelogram spanned by $\vc{a}_{,1}$ and $\vc{a}_{,2}$. Recall the notation of column and row vectors from Definition 6.2. This is visualized in Interactive Illustration 7.2. The sign of the determinant is positive (negative) if the two vectors are positively (negatively) oriented. This is illustrated with a green (red) parallelogram in the illustration. Try changing the sign of the determinant in the illustration.

For a $3 \times 3$ matrix

\begin{equation} \mx{A} = \left( \begin{array}{rr} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{array} \right) \end{equation}

(7.8)

the determinant is $\det(\mx{A}) = a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33} - a_{13} a_{22} a_{31}$. If we think of the three columns as the coordinates of three vectors in space, i.e.,

\begin{equation} \mx{A} = \left( \begin{array}{rrr} \vc{a}_{,1} & \vc{a}_{,2} & \vc{a}_{,3} \end{array} \right) \end{equation}

(7.9)

then the determinant is the volume of the parallelepiped spanned by $\vc{a}_1$, $\vc{a}_2$, and $\vc{a}_3$. In Interactive Illustration 7.3, the volume of the parallelepiped spanned by $\vc{a}_{,1}$, $\vc{a}_{,2}$, and $\vc{a}_{,3}$ is visualized.

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In the following, we relax our notation a little so that a column vector $\vc{a}_{,i}$ of a matrix $\mx{A}$ also can be denoted $\vc{a}_i$. There are several different ways one could define the determinant. What definitions one use is a technicality and the end result is the same. Here we define the determinant for a general $n \times n$ matrix $\mx{A}$ using three basic properties according to the following definition.

It turns out that there is one and only one function that fulfills these three properties. The proof of the existence and uniqueness of the determinant is a bit technical and is of less importance than the properties of the determinant. In order not to obscure the view we leave these proofs for Section 7.3. From these three basic properties follow many other useful properties which we summarize in the Theorem 7.1.

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On the surface, statement $(vi)$ can seem counterintuitive since the determinant is connected to the area or volume spanned by the column vectors, and adding something to a column vector will make it longer. How come the area stays the same even though one of the vectors become longer? The reason is illustrated in Interactive Illustration 7.4. The definitions and rules above can be used to calculate determinants. Here follows an example.

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This subsection contains the proof that there actually is a function that is fulfilling the properties in Definition 7.1, that there is only one such function, and also a formula for calculating the determinant. To complete this proof we need to introduce the notion of a permutation of $n$ numbers. The main idea of permutations is really quite simple and is best exemplified by using properties $(i)-(vi)$ on a general $2 \times 2 $ matrix.

In this example we calculated a $2 \times 2$ determinant by expanding each column. By doing this for a general $n \times n$ matrix, one obtains an explicit expression for the determinant. This expression is best expressed using the concept of a permutation of $n$ numbers.

A permutation $p$ is specified by how it acts on the set, i.e., by the numbers $p(1), p(2), \ldots, p(n)$. Since $p$ is a bijection, each number $\{ 1, 2, \ldots, n \}$ occurs precisely once in the list $p(1), p(2), \ldots, p(n)$. In other words the numbers are re-arranged or permuted. Sometimes the notation $p = [p(1), p(2), \ldots, p(n)] $ is used.

If $p$ is a permutation of $n$ elements, there are ${n \choose 2}$ pairs of numbers $(p(i),p(j))$. Some of these are in the correct order, i.e., $((p(i),p(j))$ are in the same order as $(i,j)$. If on the other hand $(p(i),p(j))$ are in the wrong order as compared to $(i,j)$, then the pair is called an inversion.

Here $\circ$ is used to denote composition, i.e., $p \circ q$ is the permutation that is the result of first permuting according to $q$ and then permuting according to $p$. (More generally composition is used for combining two functions. The composed function $f = g \circ h$ is such that $f(x) = g(h(x))$, i.e., first using function $h$ and then using $g$ on the result.) A permutation $p$ of $n$ elements can be represented as an $n \times n$ matrix $\mx{E}_p$, so that $\mx{E}_p$ is zeroes except for the elements $(p(1),1), (p(2),2) \ldots, (p(n),n)$. These elements are set to one. The determinant of $\mx{E}_p$ is particularly simple to calculate. We have $\det(\mx{E}_p) = \sigma(p)$. We are now ready to state a theorem, that gives an explicit formula for the determinant in general dimensions.

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Thus the determinants of orders $1$, $2$, and $3$ consists of $1$, $2$, and $6$ terms in accordance with the previous example. There are relatively easy patterns that one can use to remember the $2 \times 2$ determinant. Write the matrix and think about a cross. The elements on the diagonal should be multiplied and then subtract the product of the other diagonal according to the following illustration.

\begin{equation} \begin{array}{cccccc} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! + & \,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\, - \\ \!\!\!\!\!\!\!\!\!\! \searrow & \,\,\,\,\,\,\,\, \swarrow \\ a_{11} & a_{12}\\ a_{21} & a_{22} \end{array}, \end{equation}

(7.34)

i.e., $a_{11} a_{22} - a_{21} a_{12}$. For $3 \times 3$ matrices, there is also a pattern that can be used to remember the determinant. Put the columns of the matrix, $\vc{a}_1$, $\vc{a}_2$, and $\vc{a}_3$ twice, according to the following pattern,

\begin{equation} \begin{array}{cccccc} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! + &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! + &\,\,\,\,\,\,\,\,\,\,\,\, - &\,\, - & - \\ \!\!\!\!\!\!\!\!\!\!\searrow &\!\!\!\!\!\!\!\!\! \searrow &\!\!\!\!\!\!\!\!\!\! \searrow & \!\swarrow &\!\!\!\!\!\!\!\!\! \swarrow &\!\!\!\!\!\!\!\!\!\!\! \swarrow \\ a_{11} & a_{12} & a_{13} &\!\!\!\!\!\!\!\!\! a_{11} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! a_{12} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! a_{13} \\ a_{21} & a_{22} & a_{23} &\!\!\!\!\!\!\!\!\! a_{21} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! a_{22} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! a_{23} \\ a_{31} & a_{32} & a_{33} &\!\!\!\!\!\!\!\!\! a_{31} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! a_{32} &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! a_{33} \\ \end{array} \end{equation}

(7.35)

i.e., $+a_{11}a_{22}a_{33} +a_{12}a_{23}a_{31} +a_{13}a_{21}a_{32} -a_{11}a_{23}a_{32} -a_{12}a_{21}a_{33} -a_{13}a_{22}a_{31}.$ Note, however, that there is no similar simple pattern for calculating the determinant of a larger matrix, for example, there is no simple pattern for calculating the determinant of a $4 \times 4$ matrix. Note also that the number of terms grows quickly, e.g., $24$ terms for matrices of size $4 \times 4$ and $120$ terms for a $5 \times 5$ matrix.

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In this section, we present theorems about the determinant of a transposed matrix, of a product of matrices, and of the inverse of a matrix.

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Using the Theorem 7.5 we can prove the following useful result.

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In Interactive Illustration 7.5, Theorem 7.6 is shown for $2\times 2$ matrices. For example, if you make the area of the parallelogram of $\mx{A}$ smaller, then the area of the parallelogram of $\mx{A}^{-1}$ becomes larger and vice versa.

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We start this section with the definition of minor, which is a scalar value that can be computed from a square matrix.

These determinants are often called minors or first minors of the matrix.

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Expansion along a row or column is particularily useful if there are a lot of zeros in that row or column. This is illustrated in the following example

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In this section, we will introduce the adjoint matrix $\adj( \mx{A} )$ to a square matrix $\mx{A}$.

Notice that we used $D_{ji}$ and not $D_{ij}$ in the definition.

The adjoint matrix is closely related to the inverse, through the following theorem.

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In this section, Cramer's rule will be introducted. It is useful when you want the solution $\vc{x}$ to $\mx{A}\vc{x}=\vc{y}$ without needing to know the inverse matrix of $\mx{A}$.

In other words by exchanging column $i$ in $\mx{A}$ with the right hand side $\vc{y}$, and then calculating the determinant of this matrix and dividing by the determinant of $\mx{A}$, we directly obtain coordinate $i$ of the solution vector $\vc{x}$. It almost seems too easy.

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In Interactive Illustration 7.6, we solve for $\vc{z}$ in a system of equations $\mx{A}\vc{z}=\vc{b}$ using Cramer's rule.

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