Determinants and Inverses

Comp.145  October 1996  Dr. A.J. Mestel

(1) The determinant of a matrix is defined only in the case of a square matrix. It is a scalar, written detA or |A| and pronounced “det A”. For a 2 × 2 matrix,

detA = |A| = | \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} | = a_{11}a_{22} − a_{12}a_{21}.

(2) For larger order square matrices (n × n)

detA = | a_{11} ; a_{21} ; ⋮ ; a_{n1} a_{12} ; a_{22} ; ⋮ ; a_{n2} \begin{matrix}… \\ … \\ . & . & . \\ … \end{matrix} a_{1n} ; a_{2n} ; ⋮ ; a_{nn} | = ∑_{k=1}^{n} a_{ik}C_{ik} = ∑_{k=1}^{n} a_{kj}C_{kj} for any row i or column j

where C_{ij} is the cofactor (defined below) of the matrix element a_{ij}. It makes no difference which row or column is chosen. In practice one chooses one with as many zeros as possible.

(3) The cofactor C_{ij} is defined as (−1)^{i+j} times the determinant of the matrix formed by removing the i^{th} row and j^{th} column of the original matrix A. The determinant is thus defined inductively or recursively, i.e. in terms of smaller determinants. For example, a 3 × 3 determinant may be expanded along the first row to give

| \begin{matrix}a_{11} \\ a_{21} \\ a_{31} \end{matrix} \begin{matrix}a_{12} \\ a_{22} \\ a_{32} \end{matrix} \begin{matrix}a_{13} \\ a_{23} \\ a_{33} \end{matrix} | = a_{11} | \begin{matrix}a_{22} & a_{23} \\ a_{32} & a_{33} \end{matrix} | − a_{12} | \begin{matrix}a_{21} & a_{23} \\ a_{31} & a_{33} \end{matrix} | + a_{13} | \begin{matrix}a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix} | .

(4) A determinant may be positive or negative. If detA = 0 the matrix A is said to be singular or non-invertible. A determinant is zero if one of its rows is a multiple of another row. More generally, a determinant is zero if and only if the vectors forming its rows are linearly dependent i.e. one can be expressed as the sum of multiples of the others. A similar result holds for the columns.

(5) The value of a determinant is unaltered if a multiple of one row is added to another row. This fact can be used when evaluating determinants. In the following example, we subtract twice the first row from the second, then subtract five times the first row from the third, and then expand down the first column to obtain

| \begin{matrix}1 & 1 & 1 \\ 2 & 3 & 4 \\ 5 & 6 & 8 \end{matrix} | = | \begin{matrix}1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 1 & 3 \end{matrix} | = | \begin{matrix}1 & 2 \\ 1 & 3 \end{matrix} | = 1 .

It is much more efficient to evaluate large determinants this way.

(6) The determinant of a product of square matrices is equal to the product of the determinants. However, nothing useful can be said about the determinant of a sum of matrices:

det(ABD …) = detA detB detD … det(A + B) ≠ detA + detB .

(7) Suppose we have a square matrix A multiplied by a similar matrix B such that

AB = I  where  I = ( \begin{matrix}1 & 0 & … & 0 \\ 0 & 1 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 \end{matrix} )

is the identity matrix. It can be shown that BA = I also. The matrix B is called the inverse of A and is denoted A^{−1}. Clearly, (A^{−1})^{−1} = A. A matrix for which the inverse exists is said to be invertible; otherwise it is non-invertible or singular. The inverse exists if and only if detA ≠ 0.

(8) The inverse of an n×n matrix is equal to the transpose of the matrix of cofactors (defined overleaf) divided by the determinant. That is,

(A^{−1})_{ij} = \frac{1}{detA}C_{ji} .

Thus for a 2 × 2 matrix,

[ \begin{matrix}a & b \\ c & d \end{matrix} ]^{−1} = \frac{1}{ad − bc} [ \begin{matrix}d & −c \\ −b & a \end{matrix} ]^{T} = \frac{1}{ad − bc} [ \begin{matrix}d & −b \\ −c & a \end{matrix} ] .

(9) The inverse of a product of invertible matrices is equal to the product of the inverses in reverse order.

(ABD …)^{−1} = … D^{−1}B^{−1}A^{−1} .

(10) The linear equations represented by A\underline{x} = \underline{b} have a unique solution provided detA ≠ 0, namely \underline{x} = A^{−1}\underline{b}. [This is because A^{−1}\underline{b} = A^{−1}A\underline{x} = I\underline{x} = \underline{x}.] If detA = 0, then they may either have no solution or an infinite number of solutions. There are much faster and more accurate ways of solving simultaneous equations than by calculating the inverse matrix.

(11) The homogenous equations A\underline{x} = 0 have a non-zero solution only if detA = 0. Then there is generally only one solution for the ratios x_{2}/x_{1} ; x_{3}/x_{1} ; etc., giving an infinite number of solutions depending on the value of x_{1}.