Unizor - Matrices - Determinant of 3x3 Matrix - Properties

As was shown in the introductory, the following formula represents a determinant for a 3x3 matrix.
a11·a22·a33 + a21·a13·a32 + a12·a23·a31 −
− a13·a22·a31 − a21·a12·a33 − a11·a23·a32

Notice interesting properties of this formula. Each element of a matrix occurs twice in it, once in the positive member and once in a negative. Each member of a sum combines elements from different rows and different columns. Each member of a sum combines three elements of a matrix that either lie on a diagonal (main diagonal from top left corner of a matrix to a bottom right with a plus sign, the other diagonal from top right corner to bottom left - with a minus) or form a triangle in the matrix with one side parallel to the main diagonal (with a plus) or the other diagonal (with a minus).

Similarly to properties of a determinant of 2x2 matrices, a determinant of 3x3 matrices has the following basic properties.

1. If a 3x3 matrix has at least one row or one column containing all zeros (there are 3 rows and 3 columns, so we have 9 different conditions, each listing 3 elements belonging to the same row or the same column equal to zero) then its determinant equals to zero.
Indeed, for instance, the second column contains only zero elements, that is a12= 0, a22= 0 and a32= 0. Then, substituting these values into a formula for det(A), we obtain its value equal to zero.
Similarly, other cases result in exactly the same result. It would be a nice exercise to check it for all 9 cases of rows and columns.

2. If all elements of one row (or one column) of a 3x3 matrix are multiplied by some factor, the determinant will also change its value by the same factor.
Plain substitution of K·aij instead of aij into a formula for a determinant where either i is fixed on any value from 1 to 3 and j takes values 1, 2 and 3 or j is fixed on any value from 1 to 3 and i takes values 1, 2 and 3 leads to this property.

3. If one row (or one column) of a 3x3 matrix can be represented as a linear combination of two other rows (or columns) then the determinant of this matrix equals to zero. There are six different cases of such dependency for each of three rows and each of three columns, we will consider only one.
Assume that the second column equals to a linear combination of the first and the third column:
a12 = M·a11 + N·a13
a22 = M·a21 + N·a23
a32 = M·a31 + N·a33
Substituting these three values for elements of the second row into a formula for a determinant produces the following:
det(A) = a11·(M·a21+N·a23)·a33 + a21·a13·(M·a31+N·a33) +
+ a31·(M·a11+N·a13)·a23 − a11·a23·(M·a31+N·a33) −
- a21·(M·a11+N·a13)·a33 − a31·a13·(M·a21+N·a23)
Opening all parenthesis results in cancellation of all members of a sum, so the result is zero. Similarly trivial is any other case of linear dependency of one row or column from the other two. We recommend to check it as a self-study lesson.

Determinants have many other interesting properties, we will introduce them as problems later on.

[ZorShekhtman]

This lecture is a part of an educational Web site UNIZOR.COM dedicated to presentation of advanced level high school materials in different subjects, starting from Mathematics. Viewers are encouraged to go to UNIZOR.COM Web site for complete set of educational materials, including tests and exams.

[ZorShekhtman]

As was shown in the introductory, the following formula represents a determinant for a 3x3 matrix.
a11·a22·a33 + a21·a13·a32 +
+ a12·a23·a31 − a13·a22·a31 −
− a21·a12·a33 − a11·a23·a32

Notice interesting properties of this formula. Each element of a matrix occurs twice in it, once in the positive member and once in a negative. Each member of a sum combines elements from different rows and different columns. Each member of a sum combines three elements of a matrix that either lie on a diagonal (main diagonal from top left corner of a matrix to a bottom right with a plus sign, the other diagonal from top right corner to bottom left - with a minus) or form a triangle in the matrix with one side parallel to the main diagonal (with a plus) or the other diagonal (with a minus).

Similarly to properties of a determinant of 2x2 matrices, a determinant of 3x3 matrices has the following basic properties.

1. If a 3x3 matrix has at least one row or one column containing all zeros (there are 3 rows and 3 columns, so we have 9 different conditions, each listing 3 elements belonging to the same row or the same column equal to zero) then its determinant equals to zero.
Indeed, for instance, the second column contains only zero elements, that is a12= 0, a22= 0 and a32= 0. Then, substituting these values into a formula for det(A), we obtain its value equal to zero.
Similarly, other cases result in exactly the same result. It would be a nice exercise to check it for all 9 cases of rows and columns.

2. If all elements of one row (or one column) of a 3x3 matrix are multiplied by some factor, the determinant will also change its value by the same factor.
Plain substitution of K·aij instead of aij into a formula for a determinant where either i is fixed on any value from 1 to 3 and j takes values 1, 2 and 3 or j is fixed on any value from 1 to 3 and i takes values 1, 2 and 3 leads to this property.

3. If one row (or one column) of a 3x3 matrix can be represented as a linear combination of two other rows (or columns) then the determinant of this matrix equals to zero. There are six different cases of such dependency for each of three rows and each of three columns, we will consider only one.
Assume that the second column equals to a linear combination of the first and the third column:
a12 = M·a11 + N·a13
a22 = M·a21 + N·a23
a32 = M·a31 + N·a33
Substituting these three values for elements of the second row into a formula for a determinant produces the following:
det(A) =
= a11(M·a21+N·a23)a33 +
+ a21a13(M·a31+N·a33) +
+ a31(M·a11+N·a13)a23−
−a11a23(M·a31+N·a33) −
− a21(M·a11+N·a13)a33 −
− a31a13(M·a21+N·a23)
Opening all parenthesis results in cancellation of all members of a sum, so the result is zero. Similarly trivial is any other case of linear dependency of one row or column from the other two. We recommend to check it as a self-study lesson.

Determinants have many other interesting properties, we will introduce them as problems later on.