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Calculating the DeterminantFirst of all the matrix must be square (i.e. have the same number of rows as columns). Then it is just a matter of basic arithmetic. Here is how: For a 2×2 Matrix For a 2×2 matrix (2 rows and 2 columns): The determinant is: |A| = ad – bc "The determinant of A equals a times d minus b times c"
Example:
For a 3×3 Matrix For a 3×3 matrix (3 rows and 3 columns): The determinant is: |A| = a(ei - fh) - b(di - fg) + c(dh - eg) "The determinant of A equals ... etc" It may look complicated, butthere is a pattern: To work out the determinant of a 3×3 matrix:
As a formula (remember the vertical bars || mean "determinant of"):
Example:
For 4×4 Matrices and Higher The pattern continues for 4×4 matrices:
As a formula: Notice the + - + - pattern (+a... -b... +c... -d...). This is important to remember. The pattern continues for 5×5 matrices and higher. Not The Only Way This method of calculation is called the "Laplace expansion" ... I like it because the pattern is easy to remember. But there are other methods (just so you know). Summary · For a 2×2 matrix the determinant isad - bc · For a 3×3 matrix multiply a by the determinant of the 2×2 matrix that isnot in a's row or column, likewise for b and c, but remember that b has a negative sign! · The pattern continues for larger matrices: multiply aby the determinant of the matrix that isnot in a's row or column, continue like this across the whole row, but remember the + - + - pattern.
5. Systems of Linear Equations A linear equation in the variables x1, x2, . . ., xn is an equation that can be written in the form a1x1 +a2x2 +. . . anxn = b, where a1, . . ., an are the coefficients. A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables. A solution of a linear system is a list of numbers that makes each equation a true statement. The set of all possible solutions is called the solution set of the linear system. Two linear systems are called equivalent if they have the same solution set. A linear system is said to be consistent, if it has either one solution or infinitely many solutions. A system is inconsistent if it has no solutions.
Cramer’s Rule A method for solving a linear system of equations using determinants. Cramer’s rule may only be used when the system is square and the coefficient matrix isinvertible.
Date: 2015-12-11; view: 1343
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