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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 AA 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 aa, bb and cc serve as scalar multipliers to a corresponding 2-by-2 matrix.
The scalar aa is being multiplied to the 2×2 matrix of left-over elements created when vertical and horizontal line segments are drawn passing through aa.
The same process is applied to construct the 2×2 matrices for scalar multipliers bb and cc.
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.