Solving matrixes using cofactor expansion:

Co-factor Expansion: 

Formula: Given A= [aij], the (i, j) cofactor of A is the number Cij given by Cij=(-1)^(i+j) det Aij

then det A= a11C11+a12C12+......+a1nC1n. This formula is called a cofactor expansion across the first row of A. 

Theorem One: The determinant of an m x n matrix A can by computed by a cofactor expansion across any row or down any column. The expansion across the ith row using the cofactors in the above equation is det A = a11C11+a12C12+....+a1nC1n. The cofactor expansion down the jth column is det A= a1jC1j+a2jC2j+.....+anjCnj. 

The plus or minus sign is the (i,j)-cofactor depends on the position of a1j in the matrix, regardless of the sign of a1j itself. The factor (-1)^(i+j) determines the following checkerboard pattern of signs: 

Example One: Use a cofactor expansion to solve the matrix. 

Theorem One is helpful for computing the determinant of a matrix that contains many zeros. For example, if a row is mostly zeros, then the cofactor expansion across that row has many terms that are zero, and the cofactors in those terms need not to be calculated. The same approach works with a column that contains many zeros. 

Examples: Find the cofactor expansion for each matrix. 

Theorem Two: If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A. 

Example: Find the cofactor expansion

Note: Today's standards, a 25 x 25 matrix is small. Yet it would be impossible to calculate a 25 x 25 determinant by cofactor expansion. In general, a cofactor expansion requires more than n! multiplications, and 25! is approximately 1.5 x 10^25. If a computer performs one trillion multiplications per second, it would have to run for more than 500,000 years to compute a 25 x 25 determinant by this method. 

Examine Problem: Find the cofactor expansion and the determinant of the following matrix equation.