Calculating the Determinant

First 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"

It is easy to remember when you think of a cross: Blue means positive (+ad), Red means negative (-bc)

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:

• Multiply a by the determinant of the 2×2 matrix that isnot in a's row or column.
• Likewise for b, and for c
• Add them up, but remember that b has a negative sign!

As a formula (remember the vertical bars || mean "determinant of"):

"The determinant of A equals a times the determinant of ... etc"

Example:

For 4×4 Matrices and Higher

The pattern continues for 4×4 matrices:

• plus a times the determinant of the matrix that isnot in a's row or column,
• minus b times the determinant of the matrix that isnot in b's row or column,
• plus c times the determinant of the matrix that isnot in c's row or column,
• minus d times the determinant of the matrix that isnot in d's row or column,

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: 1356