Properties of determinants

Before we list the properties of determinants we give two properties of the transpose operation:

(i) $(A B)^{\top}=B^{\top} A^{\top}$, since if $A=a_{i j}$ and $B=b_{j k}$, we have

and

It is clear that the $k i$ entries are equal.

(ii) $\operatorname{det} A^{\top}=\operatorname{det} A$.

We now give properties of the determinants that allow us to make simplifications which aid in our calculations of higher order determinants. For general $n \times n$ matrices $A$ and $B$, we have:

I. $\operatorname{det} I=1$

II. $\operatorname{det}(A B)=\operatorname{det} A \operatorname{det} B$.

III. $\operatorname{det} A^{-1}=1 / \operatorname{det} A$.

This follows from properties I and II. We take the determinant of both sides of $A A^{-1}=I$ to get

IV. If we exchange two rows or two columns of a matrix, we reverse the sign of its determinant from positive to negative or from negative to positive.

$\mathrm{V}$. If we multiply one row of a matrix by a scalar $\lambda$, the determinant is multiplied by $\lambda$,

VI. The determinant behaves like a linear function on the rows of the matrix,

VII. If we have two identical rows or columns, the determinant is zero.

VIII. We may add a multiple of one row to another and this leaves the determinant unchanged. An example in two dimensions is given as follows,

where we have used properties V, VI, and VII.

IX. $\operatorname{det}(\operatorname{adj} A)=(\operatorname{det} A)^{n-1}$

To see this, we start from Cramer's rule and taking the determinant of both sides gives

where $A$ is taken to be $n \times n$ matrix. The factor $1 / \operatorname{det} A$ is a constant that multiplies each of $n$ rows therefore, using property $\mathrm{V}$, the scalar factor comes out $n$ times hence,

Finally, using property III, we have

from which the result in property IX follows.

Elementary matrices and elementary matrix operations

Many of the properties of determinants listed above can be explained in terms of elementary row and column operations. Further, elementary matrix operations are important in finding matrix inverses as well as solving simultaneous linear equations.

The Elementary matrix

An elementary matrix is an $n \times n$ matrix which can be obtained from the identity matrix $I_{n}$ by performing on $I_{n}$ a single elementary row transformation. For example,

is an elementary matrix obtained by switching rows 1 and 2 of the identity matrix $\left(R_{1}\right) \leftrightarrow\left(R_{2}\right)$

There are three types of elementary matrix operations:

1. Interchange two rows (or columns);
2. Multiply each element in a row (or column) by a scalar;
3. Adding a multiple of one row (or column) to another.

Since there are three elementary transformations, there are three kinds of elementary matrices (one was shown in Definition 10.4 above). Elementary matrices are used to simulate elementary row (or column) transformations. To perform an elementary transformation on a matrix $A$ we pre-multiply $A$ by the elementary matrix obtained by applying the same transformation to the identity matrix. Next, we consider each one of the three types of matrix operations as they apply to rows. We apply all row transformations on the following matrix $A$

I. Interchange rows

Suppose we want to interchange rows 1 and 3 of the following matrix, We start with the $3 \times 3$ identity matrix and interchange rows 1 and 3 to obtain

where $E$ is the desired elementary matrix; we can check that if we pre-multiply $A$ by this matrix, the resulting matrix is $A$ with rows 1 and 3 interchanged,

Note that to interchange columns i.e. $\left(C_{i}\right) \leftrightarrow\left(C_{j}\right)$, we post-multiply $A$ by $E$.

II. Multiply each element in a row by a scalar

Suppose we want to multiply row 2 of matrix $A$ by $\lambda$, i.e. $\left(\lambda R_{2}\right) \rightarrow\left(R_{2}\right)$. To create the elementary row matrix we multiply each element in the second row of the identity matrix by $\lambda$,

Then we pre-multiply $A$ by $E$ to multiply each element in the second row of $A$ by $\lambda$,

III. Adding a multiple of one row to another.

Finally, suppose we want to perform the operation $\left(R_{3}-2 R_{1}\right) \rightarrow\left(R_{3}\right)$ i.e. we replace row 3 by the sum of row 3 to -2 times row 1 . We perform this transformation on the identity matrix yielding the elementary matrix,

We then pre-multiply $A$ by $E$ to carry out the row transformation,

All of the above transformations can be repeated for column operations; recall that in order to perform operation the matrix $A$ is post-multiplied by $E$.

Properties of elementary matrices

I. The inverse of the elementary matrix that interchanges two rows is itself. For instance, the inverse of

II. The inverse of the elementary matrix that represents $\left(\lambda R_{i}\right) \rightarrow\left(R_{i}\right)$ is the elementary matrix that represents $\left(\lambda^{-1} R_{i}\right) \rightarrow R_{i}$. For example, the inverse of

III. The inverse of the elementary matrix that gives $\left(R_{j}+\lambda R_{i}\right) \rightarrow\left(R_{j}\right)$ is the elementary matrix which represents $\left(R_{j}-\lambda R_{i}\right) \rightarrow\left(R_{j}\right)$. Next, we carry out an example with elementary matrix operations to facilitate the calculation of a $4 \times 4$ determinant.

Example 10.7 Calculate the determinant of

Solution We have already seen in Example 10.6 that calculating determinants of high dimensional matrices is quite tedious. Here, we perform row and column operations to simplify the calculation of the determinant.

We first perform the row operation $\left(R_{1}-R_{2}\right) \rightarrow\left(R_{1}\right)$ since this will give us two zeros in the first row,

A third zero in the first row can be obtained by performing the column operation $\left(C_{4}-C_{3}\right) \rightarrow\left(C_{4}\right)$

Now, taking the determinant by expansion along the first row, we obtain

note that we have only added/subtracted rows/columns from other rows/columns in $A$ in our matrix operations and according to property VIII (see Section 10.3) these leave the determinant unchanged. Hence the determinant in Eq. (10.28) is equivalent to the determinant of $A$. We can see that it is possible to simplify this further by obtaining more zeros in the first column of the determinant given in (10.28). Carrying out $\left(R_{2}-R_{1}\right) \rightarrow\left(R_{2}\right)$ gives

followed by $\left(R_{3}-4 R_{1}\right) \rightarrow\left(R_{3}\right)$,

Exercises

1. Use row and column operations to determine the following determinants:

   (a) $\left|\begin{array}{cccc}1 & -1 & 2 & 3 \\ 2 & -2 & 6 & 3 \\ -1 & 3 & 2 & 1 \\ 3 & 1 & 3 & 1\end{array}\right|$

   (b) $\left|\begin{array}{cccc}1 & 0 & 2 & 1 \\ 1 & 1 & 4 & 3 \\ -1 & -1 & 1 & -1 \\ -2 & 2 & 3 & -3\end{array}\right|$.

2. Given the $3 \times 3$ matrix that adds 3 times the third row to the first row, should we pre- or post-multiply by this matrix?