# 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
(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):

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:

 |B| = 4×8 - 6×3 = 32-18 = 14

## 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, but there is a pattern:

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"):

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

### Example:

 |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,

As a formula:

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.