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Determinants Section
The focus before was to determine information about the solution set of the
linear system of equations given as the matrix equation Ax = b. We saw that in
general, both the coefficient matrix A and right side b contributed to the specific
nature of the solution set. This followed since the linear system Ax = b is
consistent if and only if b is a linear combination of the columns of A. A goal was
to determine the coefficients of such linear combinations and to do so we used
row operations. However, in the special case of a nonsingular matrix A it was an
intrinsic quality of the matrix A that the linear system Ax = b was consistent
regardless of the contents of b.
Here we develop an intrinsic number associated with a square matrix. This
intrinsic number is called the determinant of the square matrix.
• We make a formal definition for the determinant of 2 2 and 3  3 matrices.
• We show how to generalize the determinant concept to n  n matrices.
• We develop properties of the determinant and show how it is related to linear
systems that have a unique solution.
• We show how to use row operations to efficiently compute the determinant.
The determinant of n × n matrix A, denoted det(A), is a function which associates
with matrix A a unique numerical value. (The determinant is defined only for square
matrices.)
det(A) = numerical value
Computing the Determinant of 2 × 2 and 3 × 3 Matrices
Rather than compute det(A) using the expression given above we present the
following scheme for generating the products required. Adjoin to A its first two
columns, then compute the products along the lines shown in Figure 1. The
products are then combined with the sign that appears at the end of the line. Those
products obtained along the lines parallel to the diagonal have a plus sign associated
with them and those not parallel to the diagonal have a minus sign.
Figure 1.
The computational procedure shown in Figure 1 is called the '3  3 device' for the
determinant of a 3  3 matrix.
Warning: there are no such short cuts or 'devices' for larger matrices.
Figure 2.
det(A) = (-2) + (2) + 0
-0-
(-8) - (3)
= 5
We have devices for computing the determinants of 2  2 and 3  3 matrices.
How do we compute determinants of larger matrices?
We discuss two techniques.
The first technique expresses the determinant of a matrix A as a linear combination of
determinants of submatrices of A. This called determinants by expansion.
The second technique uses row operations on A to obtain an upper triangular matrix
while we keep track of how the row operations used effect the value of the
determinant.
Determinants by Expansion
Here we develop a procedure for computing the determinant of an n × n matrix as a
linear combination of determinants of submatrices matrices. We call this method
determinants by expansion; it is also called Cofactor Expansion or Laplace
Expansion by Cofactors.
The expression for the determinant of a 3  3 matrix can be written as an expansion
involving determinants of 2  2 submatrices and the entries from the first row. We
have (verify by using the 2  2 device and simplifying)
Note the use of the entries
of row #1 as coefficients
with signs attached.
To get the signs attached to the coefficients we use incorporate the row and column
numbers of the rows and columns omitted by writing the expression in the form
(1,1) entry = a
(1,2) entry = b
(1,3) entry = c
We call this expression the expansion of the determinant about the first row.
In fact, we can use any row or column for the expansion with appropriate
powers of (-1) multiplying the entries and submatrices selected by omitting a
row and column.
Sign pattern for
expansion method
for a 3 × 3 matrix.
The definition we give for the determinant of an n  n matrix is an expansion about the
first row following the pattern of the 3  3 case as given above. However, the
expression involves determinants of (n -1)  (n -1) matrices, hence is recursive.
Suppose we have two matrices one is n × n matrix A and the other is n × n matrix
B, we have
det(AB) = det(A) det(B).
Using magnification factor ideas we know that when A = identity matrix the
magnification factor is 1 since the image is exactly the same as the original figure.
Suppose A is nonsingular then we know A A-1 = I. Thus we have
det(A A-1) = det(A) det(A-1) = det(I) = 1.
For A nonsingular we have:
Since a matrix is either singular or nonsingular we also have
det(singular matrix) = 0.
Summary of the properties of the determinant
Computing the Determinant using Row Operations
Computing determinant of an n × n matrix for large n using expansion is quite tedious. A
more efficient technique involves the use of row operations. However, row operations
can change the value of a determinant. The following properties can be proven; we will
assume they are true.:
and det(upper triangular matrix) = product of its diagonal entries.
Using these four properties we will formulate a procedure for computing the
determinant of any square matrix by using row operations to reduce it to upper
triangular form and keeping track of how the row operations employed effect the
determinant.
det of original matrix:
Another example: Compute the determinant of
Ans = 184