Applied Discrete Structures

Associated with every square matrix is a number called its determinant. The most important information it provides us with is whether the matrix is invertible. A matrix has an inverse if and only if its determinant is nonzero. If $A$ is a square matrix, then the determinant of $A$ is commonly denoted either $det(A)$ or $|A|\text{.}$ Strictly speaking, we only need to define the determinant of a $1\times 1$ matrix here and then define the higher ordered ones recursively, but for convenience we also recall the definition of the determinant of a $2\times 2$ matrix.

We now proceed to define the determinant of an $n\times n$ matrix where $n>2\text{.}$ This definition requires two preliminary definitions those of minors and cofactors.

Finally, we will define the determinant of a square matrix. Our definition is practical in that you can apply it easily to any matrix. It isn’t the most general, nor is it the best definiton for the purposes of proving properties of determinants. The more general definition is beyond our current scope, but can be easily stated with background in permutation groups.

Our definition of a determinant involves what is called expansion along the first row of the matrix A. It is certainly not obvious, but it is true, that the determinant of a matrix can be found by expanding along any row or any column.