Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from a square matrix.

A Matrix is an array of numbers:

A Matrix
A Matrix
(This one has 2 Rows and 2 Columns)

The determinant of that matrix is (calculations are explained later):

3×6 − 8×4 = 18 − 32 = −14

What is it for?

The determinant tells us things about the matrix that are useful in systems of linear equations, helps us find the inverse of a matrix, is useful in calculus and more.

Symbol

The symbol for determinant is two vertical lines either side.

Example:

|A| means the determinant of the matrix A

(Exactly the same symbol as absolute value.)

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 basic arithmetic. Here is how:

For a 2×2 Matrix

For a 2×2 matrix (2 rows and 2 columns):

A Matrix

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)
  A Matrix

Example:

A Matrix

|B| = 4×8 - 6×3
  = 32-18
  = 14

 

For a 3×3 Matrix

For a 3×3 matrix (3 rows and 3 columns):

A Matrix

The determinant is:

|A| = a(ei - fh) - b(di - fg) + c(dh - eg)
"The determinant of A equals ... etc"

It may look complicated, but there is a pattern:

A Matrix

To work out the determinant of a 3×3 matrix:

  • Multiply a by the determinant of the 2×2 matrix that is not 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"):

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

Example:

A Matrix

|C| = 6×(-2×7 - 5×8) - 1×(4×7 - 5×2) + 1×(4×8 - -2×2)
  = 6×(-54) - 1×(18) + 1×(36)
  = -306

For 4×4 Matrices and Higher

The pattern continues for 4×4 matrices:

  • plus a times the determinant of the matrix that is not in a's row or column,
  • minus b times the determinant of the matrix that is not in b's row or column,
  • plus c times the determinant of the matrix that is not in c's row or column,
  • minus d times the determinant of the matrix that is not in d's row or column,

A Matrix

As a formula:

A Matrix

Notice the + - + - pattern (+a... -b... +c... -d...). This is important to remember.

 

The pattern continues for 5×5 matrices and higher. Usually best to use a Matrix Calculator for those!

 

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 is ad - bc
  • For a 3×3 matrix multiply a by the determinant of the 2×2 matrix that is not 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 a by the determinant of the matrix that is not in a's row or column, continue like this across the whole row, but remember the + - + - pattern.