The Formula of the Determinant of 3×3 Matrix

The standard formula to find the determinant of a 3×3 matrix is a break down of smaller 2×2 determinant problems which are very easy to handle. If you need a refresher, check out my other lesson on how to find the determinant of a 2×2. Suppose we are given a square matrix A where,

Matrix A is a square matrix with a dimension of 3x3 wherein the first row contains the elements a,b, and c; the second row contains the elements d, e, and f; and finally, the third row contains in the entries g, h, and i. In short form, matrix A can be expressed as A = [a,b,c;d,e,f;g,h,i].

The determinant of matrix A is calculated as

The determinant of matrix A = [a,b,c;d,e,f;g,h,i] is calculated as determinant of A = det(A) = det [a,b,c;d,e,f;g,h,i] = a times determinant of matrix [e,f;h,] minus b times determinant of matrix [d,f;g,i] + c times determinant of [d,e;g,h].

Here are the key points:

  • Notice that the top row elements namely a, b and c serve as scalar multipliers to a corresponding 2-by-2 matrix.
  • The scalar a is being multiplied to the 2×2 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 2×2 matrices for scalar multipliers b and c.

Determinant of 3 x 3 Matrix (animated)

This is an animated GIF file that shows the step-by-step procedure how to find the determinant of a 3 by 3 matrix with entries a, b, and c on its first row; entries d, e and f on its second row; and entries g, h, and i on its third row. The formula is det(A) = det[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].

Examples of How to Find the Determinant of a 3×3 Matrix

Example 1: Find the determinant of the 3×3 matrix below.

This is a 3x3 square matrix that has the following elements on the first row, second row, and third row, respectively; 2,-3, and 1; 2, 0, and -1; 1, 4 and 5. In compact form, we can write this as [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 3x3 matrix with elements [a,b,c;d,e,f;g,h,i] is equal to the 3 by 3 matrix with elements [2,-3,1;2,0,-1;1,4,5]

Applying the formula,

the formula to find the determinant of a square matrix (3x3) is determinant of [a,b,c;d,e,f;g,h,i] = a times the determinant of [e,f;h,i] minus b times the determinant of [d,f;g,i] plus the c times the determinant of [d,e;g,h]
the determinant of matrix [2,-3,1;2,0,-1;1,4,5] is calculated as 2 times the determinant of [0,-1;4,5] minus (-3) times the determinant of [2,-1;1,5] plus 1 times the determinant of [2,0;1,4] which can be further simplified as 2+3+1[8-0]= 2 (0+4) +3 (10+1) + 1 (8-0) = 2(4)+3(11)+1(8)=8+33+8=49, therefore det[2,-3,1;2,0,-1;1,4,5] = 49

Example 2: Evaluate the determinant of the 3×3 matrix below.

this is a square matrix with 3 rows and 3 columns, that is a square matrix with a size of 3 x 3. it has entries of 1,3, and 2 on its first row; entries of -3,-1 and -3 on its second row; and entries 2,3 and 1 on its third row. in short format, we can rewrite this as [1,3,2;-3,-1,-4;-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,

matrix [a,b,c; d,e,f; g,h,i] is equal to matrix [1,3,2;-3,-1,-3;2,3,1]

our calculation of the determinant becomes…

determinant of [a,b,c;d,e,f;g,h,i] = a * determinant of [e,f;h,i] - b * determinant of [d,f;g,i] + c * determinant of [d,e;g,h]
det [1,3,2;-3,-1,-3;2,3,1] = 1 * det [-1,-3;3,1] - 3 * det [-3,-3;2,1] + 2 det [-3,-1;2,3] = 1*[-1-(-9)]-3*[-3-(-6)]+2 *[-9-(-2)] = 1(8) -3(3)+2(-7) = 8-9-14 = -15

Example 3: Solve for the determinant of the 3×3 matrix below.

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 in the entire expression to disappear.

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

this is a 3x3 square matrix with elements -5, 0 and -1 on the first row; elements 1,2 and -1 on the second row; and elements -3,4 and 1 on the third row

Using the formula, we have…

the formula to calculate or compute for the determinant of a 3x3 matrix is det[a,b,c;d,e,f;g,h,i] = a*det[e,f;h,i]-b*det[d,f;g,i]+c*[d,e;g,h]
the determinant of the square matrix [-5,0,1;1,2,-1;-3,4,1] equals 5 times the determinant of [2,-1;4,1] minus 0 times the determinant of [1,-1;-3,1] plus (-1) times the determinant of [1,2;-3,4] = 5 - 0 - [4 - (-6)] = -5 (2+4) -0 (1-3) - 1(4+6) = -5(6)-2(-2)-1(10)=-30-0-10 = -40. The final answer is determinant of [-5,0,1;1,2,-1;-3,4,1] = -40

Practice with Worksheets

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Determinants of a 2×2 Matrix