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Related Lessons: Determinant of 2x2 Matrix

 

Determinant of 3x3 Matrix | The Standard Formula

 

The standard formula to find the determinant of a 3x3 matrix is a break down of smaller 2x2 determinant problems which are very easy to handle. If you need a refresher how to find the determinant of a 2x2, check it out on my separate lesson.

 

Suppose we are given a square matrix A where,

A = [ [a b c] [d e f] [g h i] ]

The determinant of matrix A is calculated as

image showing the formula of determinant of 3x3 matrix: det (A) = det A = [ [a b c] [d e f] [g h i] ] = a*det[[e f][h i]] - b* det[ [d f] [g i] ] + c* det[ [d e] [g h] ]

 

Below is the formula in animated format which could hopefully help you remember it. Here are the key points:

  • Notice that the top row elements namely ab and c serve as scalar multipliers to a corresponding 2-by-2 matrix.
  • The scalar a is being multiplied to the 2x2 matrix of left-over elements created when vertical and horizontal line segments are drawn passing through a.
  • The same process is applied to construct the 2x2 matrices for scalar multipliers b and c.

 

Determinant of 3 x 3 Matrix (animated)
animated image demonstrating the formula of the determinant of a 3x3 matrix

 


 

Example 1: Find the determinant of the 3x3 matrix [ [2 -3 1] [2 0 -1] [1 4 5] ].

The set-up below will help you find the correspondence between the generic elements of the formula and the elements of the actual problem.

[ [a b c] [d e f] [g h i] ] = [ [2 -3 1] [2 0 -1] [1 4 5] ]

Applying the formula...

formula to find the determinant of 3x3 matrix

 

det [ [2 -3 1] [2 0 -1] [1 4 5] ] = 49

 


 

Example 2: Evaluate the determinant of the 3x3 matrix [ [1 3 2] [-3 -1 -3] [2 3 1] ].

Be very careful when substituting the values into the right places in the formula. Common errors occur when students become careless during the initial step of substitution of values.

In addition, take your time to make sure your arithmetic is also correct. Otherwise, a single error somewhere in the calculation will yield a wrong answer in the end.

Since,

[ [a b c] [d e f] [g h i] ] =  [ [1 3 2] [-3 -1 -3] [2 3 1] ]

our calculation of the determinant becomes...

standard formula of the determinant of 3x3 matrix

 

deteminant of matrix [ [1 3 2] [-3 -1 -3] [2 3 1] ] = -15

 

 


 

Example 3: Solve for the determinant of the 3x3 matrix [ [-5 0 -1] [1 2 -1] [-3 4 1]].

The presence of zero (0) in the first row should make our computation much easier. Remember, those elements in the first row act as scalar multipliers. Therefore, zero multiplied to anything will result to the entire expression to disappear.

Here's the setup again to show the corresponding numerical value of each variable in the formula.

[ [a b c] [d e f] [g h i] ] =  [ [-5 0 -1] [1 2 -1] [-3 4 1]]

Using the formula, we have...

expanded formula of det of 3x3 matrix

 

determinant of matrix [ [-5 0 -1] [1 2 -1] [-3 4 1]] = -40

 

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

 

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