Let A=(abbc) be a 2×2 matrix. If ad=bc=0, then A is invertible. Further, if ad−bc=0, then A is not invertible.
This means that there is one number associated with each 2×2 matrix that determines whether it is invertible or not. This number turns out to exist for any n×n matrix as well.
For any n×n square matrix A, let Aij denote the n−1×n−1 submatrix of A obtained by removing the ith row and the jth column of A.
Ex.
AA12=⎝⎜⎜⎛4234122431157437⎠⎟⎟⎞=⎝⎛234115437⎠⎞
For n>2, the determinant of an n×n matrix A whose entries are denoted as aij is given by
detA=a11detA11−a12detA12+...+(−1)n+1a1ndetA1n, or, more compactly,
detA=j=1∑n (−1)1+j a1j detA1j.
Ex. A=⎝⎛12054−20−10⎠⎞
detA=(−1)2 a11 detA11+(−1)3 a12 detA12+(−1)4 a13 detA13=1 det(4−2−10)−5 det(20−10)+0 det(204−2)=1(0−2)−5(0−0)+0=−2.
Given an n×n matrix A, let Cij=(−1)i+j detAij. The determinant of A now becomes
detA=a11C11+a12C12+...+a1nC1n. These C-matrices are known as cofactors of A.
Theorem: The determinant of a n×n matrix A can be computed by a cofactor expansion
detA=aj1Cj1+aj2Cj2+...+ajnCjn
or
detA=a1jC1j+a2jC2j+...+anjCnj.
This means the determinant can be computed across any row or column, rather than just the first row as shown previously. Because of this, we can speed up the expensive computation required to find the determinant by computing across a row or column with many zeroes.
Ex. We can more quickly find the determinant of the matrix from the previous example by computing along the third row instead of the first. A=⎝⎛12054−20−10⎠⎞
detA=(−1)4 a31 detA31+(−1)5 a32 detA32+(−1)6 a33 detA33=0 det(540−1)+2 det(120−1)+0 det(1254)=0+2(−1−0)+0=−2.
Definition. A diagonal matrix is a matrix where all of the non-diagonal entries are zero.
For a diagonal matrix A, the determinant of A is the product of its diagonal entries.
Ex.
AdetA=⎝⎜⎜⎜⎜⎛1000002000003000004000005⎠⎟⎟⎟⎟⎞=(1)(2)(3)(4)(5)=120.