Determinants

The determinant of a matrix ${\bf A}$ is denoted $\vert {\bf A} \vert$ and is a scalar quantity (i.e., a number). This number is involved in computation of inverse matrices (below). For the trivial case of a 1x1 matrix, the determinant is just the number in the matrix. For a 2x2 matrix, the determinant is easily computed as

$$\left\vert \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right\vert = a_{11} a_{22} - a_{21} a_{12}.$$ (19)

For a 3x3 matrix, the determinant is again easily computed, being

$$\left\vert \begin{array}{ccc} a_{11} a_{12} a_{13} \\ a_... ...} a_{22} a_{13} - a_{32} a_{23} a_{11} - a_{33} a_{21} a_{12}.$$ (20)

Note that each term consists of the product of three factors. The positive terms can be obtained by starting on the top row and multiplying by elements going down and to the right diagonally, wrapping around when hitting the edge of the determinant. Similarly, negative terms can be obtained by starting on the bottom row and multiplying by elements going up and to the right diagonally, wrapping around when hitting an edge.

Another simple case for computing determinants is that of diagonal (or triangular) matrices, where the determinant is just the product of the entries on the main diagonal.

For more general matrices, there is a procedure for computing determinants which involves cofactor matrices

$$\vert{\bf A}\vert = a_{i1} A_{i1} + a_{i2} A_{i2} + \cdots + a_{in} A_{in},$$ (21)

where $a_{ij}$ are the elements of matrix ${\bf A}$ and $A_{ij}$ are the cofactors, which are $(-1)^{i+j}$ times the determinant of a smaller $(n-1) \times (n-1)$ matrix found by eliminating row $i$ and column $j$ [ $A_{ij} = (-1)^{i+j} \vert{\bf M}_{ij}\vert$ ]. Obviously there is a problem if we are defining a determinant in terms of other determinants! However, we can apply these rules iteratively until we get to 3x3 or 2x2 matrices, for which we can take determinants using the simple rules given above. You may wish to confirm that the cofactor method, when applied to the case of a 3x3 matrix, yields equation (20) when the cofactors of the smaller 2x2 matrices are evaluated using the aid of equation (19).