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Co-factoring the determinant of a 3x3 matrix
The definition of the determinant of a 3x3 matrix was introduced in the lesson Determinant of a 3x3 matrix under the current topic in this site.
According to the definition, the determinant of a 3x3 matrix is expressed by a quite long formula.
Co-factoring the determinant is expanding of it into the (alternate) sum of products of its elements and the determinants of smaller 2x2 complementary sub-matrices.
Co-factoring explains how the determinant of a nxn matrix is connected with the determinants of smaller sub-matrixes. Sometimes co-factoring facilitates calculations of determinants. Let us start with reminding the definition of the determinant of a 3x3 matrix from the referred lesson.
Definition
Let us consider a 3x3-matrix =
comprised of numbers , , , , , , , , and .
The determinant of the matrix =
is the number = + + - - - .
Thus the determinant is defined for any square matrix with 3 rows and 3 columns.
Sometimes the determinant of a matrix
is denoted as det
or
.
So, det
=
= + + - - - .
Notice the patterns for calculation the determinant terms in Figure 1a and 1b. .
The major matrix diagonal and two triangles in the Figure 1a that have the sides parallel to the major matrix
diagonal each contributes the product of three matrix elements with the sign "as is" to the determinant.
The auxiliary diagonal and two triangles in the Figure 1b that have the sides parallel to the auxiliary diagonal
each contributes the product of three matrix elements with the opposite sign to the determinant.
Fig. 1a. (+)-Patterns
for the determinant
Fig. 1b. (-)-Patterns
for the determinant
Figures 2a - 2f represent each term of the determinant visually.
Also notice that each term of a determinant (additive or subtrahend) is the product of elements that are located in different rows and different columns of a matrix.
No one single term of a determinant contains two elements from the same row. No one single term of a determinant contains two elements from the same column.
Fig. 2a. The term
Fig. 2b. The term
Fig. 2c. The term
Fig. 2d. The term
Fig. 2e. The term
Fig. 2f. The term
Theorem (co-factoring determinant along the row)
The determinant of a 3x3 matrix =
is equal to = - + ,
where , and are complementary 2x2 sub-matrices to the elements , and in the matrix , respectively.
Complementary sub-matrix to an element of a matrix is
the matrix which is obtained from the original matrix after deleting
the i-th row and j-th column where the element is located.
Figures 3a, 3b and 3c show the complementary sub-matrices to the
elements , and of the first row of the 3x3 matrix .
Figure 3a. Complementary
sub-matrix
Figure 3b. Complementary
sub-matrix
Figure 3c. Complementary
sub-matrix
Proof
The proof is straight-forward. Simply re-group the terms in the determinant definition by grouping the pairs of additives that have the common factors , and .
det
= + + - - - =
= ( - ) + ( - ) + ( - ) =
= .( - ) - .( - ) + .( - ) = - + .
It is what has to be proved.
The same proof is shown in Figures 4a - 4c visually.
Combination of Fig.2a and Fig.2f
Fig. 4a. The terms -
=.()
Combination of Fig.2b and Fig.2e
Fig. 4b. The terms -
=-.()
Combination of Fig.2c and Fig.2d
Fig. 4c. The terms )
= .()
Expansion in the Theorem is co-factoring the determinant along the first row. Similarly to that there are formulas for co-factoring the determinant along other rows.
For example, co-factoring along the second row is = - + - (notice the signs in this expansion).
Co-factoring along the third row is = - + (again, notice the signs).
Here are complementary 2x2 sub-matrices to the elements in the matrix .
The rule of signs in co-factoring the 3x3 determinants is given by the matrix in the Figure 5.
These are the signs to put them before each expansion term.
There are formulas for co-factoring the determinant along the columns.
For example, co-factoring along the first column is = - + .
Co-factoring along the second column is = - + - .
Co-factoring along the third column is = - + .
Again, the signs before the individual terms are ruled by the matrix in the Figure 5.
