Lecture 8

The Determinant
There is another way to solve systems of equations with three variables. It involves a quantity called the determinant.
Every m×m matrix has a unique determinant. The determinant is a single number. To find the determinant of a 2×2matrix , multiply the numbers on the downward diagonal and subtract the product of the numbers on the upward diagonal:
A =



detA = a 1 b 2 - a 2 b 1 .
For example,
det = 4(6) - (- 1)(- 2) = 24 - 2 = 22



To find the determinant of a 3×3 matrix, copy the first two columns of the matrix to the right of the original matrix. Next, multiply the numbers on the three downward diagonals, and add these products together. Multiply the numbers on the upward diagonals, and add these products together. Then subtract the sum of the products of the upward diagonals from the sum of the product of the downward diagonals (subtract the second number from the first number):
in linear algebra, Cramer s rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750,[1] although Colin Maclaurin also published special cases of the rule in 1748 (and possibly knew of it as early as 1729).