Análisis de resultados de las entrevistas a los actores clave

Recall the three row operations we used for the Gauss–Jordan elimi-nation procedure:

• Add a multiple of one row to another row

• Swap two rows

• Multiply a row by a constant

We’ll now describe the effects of these row operations on the value of the matrix determinant. In each case, we’ll connect the effects of the row operation to the geometrical interpretation of the determinant operation.

Add a multiple of one row to another row

Adding a multiple of one row of a matrix to another row does not change the determinant of the matrix.

Figure 4.8: Row operations of the form R^α : Ri ← Rⁱ+ αRj do not change the value of the matrix determinant.

This property follows from the fact that parallelepipeds with equal base enclosed between two parallel planes have the same volume even if they have different slants. This is known as Cavalieri’s principle.

It is easier to visualize Cavalieri’s principle in two dimensions by considering two parallelograms with base b and different slants, en-closed between two parallel lines. The area of both parallelograms is the same A = b×h, where h is the distance between the parallel lines.

Swap rows

Swapping two rows of a matrix changes the sign of its determinant.

Figure 4.9: Row-swaps, Rβ: Ri↔ R^j, flip the sign of the determinant.

This property is a consequence of measuring signed volumes. Swap-ping two rows changes the relative orientation of the vectors, and hence the sign of the volume.

Multiply a row by a constant

Multiplying a row by a constant is equivalent to the constant multi-plying the determinant.

Figure 4.10: Row operations of the form R^γ : Ri← αRⁱscale the value of the determinant by the factor α.

The third property follows from the fact that making one side of the parallelepiped α times longer increases the volume of the paral-lelepiped by a factor of α.

When each entry of an n×n matrix is multiplied by the constant α, each of the n rows is multiplied by α so the determinant changes by a factor of αⁿ: det(αA) = αⁿdet(A).

Zero-vs-nonzero determinant property

There is an important distinction between matrices with zero deter-minant and matrices with nonzero deterdeter-minant. We can understand this distinction geometrically by considering the 3 × 3 determinant calculation. Recall, the volume of the parallelepiped with sides ~u,

~v, and ~w is equal to the determinant of the matrix containing the vectors ~u, ~v, and ~w as rows. If the determinant is zero, it means at least one of the rows of the matrix is a linear combination of the other rows. The volume associated with this determinant is zero because the geometrical shape it corresponds to is a flattened, two-dimensional parallelepiped, in other words, a parallelogram. We say the matrix is

“deficient” if its determinant is zero.

4.4 DETERMINANTS 165

On the other hand, if the determinant of a matrix is nonzero, the rows of the matrix are linearly independent. In this case, the deter-minant calculation corresponds to the volume of a real parallelepiped.

We say the matrix is “full” if its determinant is nonzero.

The zero-vs-nonzero determinant property of a matrix does not change when we perform row operations on the matrix. If a matrix A has a nonzero determinant, we know its reduced row echelon form will also have nonzero determinant. The number of nonzero rows in the reduced row echelon form of the matrix is called the rank of the matrix. We say a matrix A ∈ R^n×nhas full rank if its RREF contains npivots. If the RREF of the matrix A contains a row of zeros, then Ais not full rank and det(A) = 0. On the other hand, if det(A) 6= 0, we know that rref(A) = 1.

Applications

Apart from the geometric and invertibility-testing applications of de-terminants described above, dede-terminants are related to many other topics in linear algebra. We’ll briefly cover some of these below.

Cross product as a determinant

We can compute the cross product of the vectors ~v = (v1, v2, v3)and

~

w = (w1, w2, w3) by computing the determinant of a special matrix.

We place the symbols ˆı, ˆ, and ˆk in the first row of the matrix, then write the coefficients of ~v and ~w in the second and third rows. After expanding the determinant along the first row, we obtain the cross product:

Observe that the anti-linear property of the vector cross product

~v× ~w =− ~w× ~v corresponds to the swapping-rows-changes-the-sign property of determinants.

The extended-array trick for computing 3 × 3 determinants (see Figure 4.5) doubles as a trick for computing cross-products:

Figure 4.11: We can quickly compute the cross product of two vectors using the extended-array trick.

Using the correspondence between the cross-product and the deter-minant, we can write the determinant of a 3 × 3 matrix in terms of the dot product and cross product:

u1 u2 u3

v1 v2 v3

w1 w2 w3

= ~u· (~v × ~w).

Cramer’s rule

Cramer’s rule is an approach for solving systems of linear equations using determinants. Consider the following system of equations and its representation as a matrix equation:

a11x1+ a12x2+ a13x3= b1

a21x1+ a22x2+ a23x3= b2 ⇔ A~x = ~b.

a31x1+ a32x2+ a33x3= b3

We’re looking for the vector ~x = (x1, x2, x3)that satisfies this system of equations.

Begin by writing the system of equations as an augmented matrix:





a11 a12 a13 b1

a21 a22 a23 b2

a31 a32 a33 b3



 ≡





| | | |

~a1 ~a2 ~a3 ~b

| | | |



.

We use the notation ~aj to denote the j^thcolumn of coefficients in the matrix A, and ~b to denote the column of constants.

Cramer’s rule requires computing ratios of determinants. To find x1, the first component of the solution vector ~x, we compute the following

4.4 DETERMINANTS 167

Basically, we replace the column that corresponds to the unknown we want to solve for (in this case the first column) with the vector of constants ~b, and compute the determinant before dividing by |A|. To find x2, we’ll need to compute the determinant of a matrix where ~b replaces the second column of A. Similarly, to find x3, we replace the third column with ~b.

Cramer’s rule is a neat computational trick that might come in handy if you ever want to solve for one particular coefficient in the unknown vector ~x, without solving for the other coefficients.

Linear independence test

Suppose you’re given a set of n, n-dimensional vectors {~v1, ~v2, . . . , ~vn} and asked to check whether the vectors are linearly independent.

You could use the Gauss–Jordan elimination procedure to accom-plish this task. Write the vectors ~vias the rows of a matrix M. Next, use row operations to find the reduced row echelon form (RREF) of the matrix M. Row operations do not change the linear independence between the rows of a matrix, so you can tell whether the rows are independent from the reduced row echelon form of the matrix M.

Alternatively, you can use the determinant test as a shortcut to check whether the vectors are linearly independent. If det(M) is zero, the vectors that form the rows of M are not linearly independent. On the other hand, if det(M) 6= 0, then the rows of M and linearly independent.

Eigenvalues

The determinant operation is used to define the characteristic poly-nomial of a matrix. The characteristic polypoly-nomial of A is

pA(λ)≡ det(A − λ1)

The roots of the characteristic polynomial are the eigenvalues of the matrix A. Observe the coefficient of the linear term in pA(λ)is equal to −Tr(A) and the constant term equals det(A). The name char-acteristic polynomial is indeed appropriate since pA(λ) encodes the information about three important properties of the matrix A: its eigenvalues (λ1, λ2), its trace Tr(A), and its determinant det(A).

At this point, we don’t need to delve into a detailed discussion about properties of the characteristic polynomial. We gave the def-inition of pA(λ) here because it involves the determinant, and we’re in the section on determinants. Specifically, pA(λ) is defined as the determinant of A with λs (the Greek letter lambda) subtracted from the entries on the diagonal of A. We’ll continue the discussion on the characteristic polynomial and eigenvalues in Section 7.1.

Exercises

E4.6 Find the determinant of the following matrices:

A =

Observe that the matrix B can be obtained from the matrix A by swapping the first and second rows of the matrix. We therefore expect det(A)and det(B) to have the same absolute value but opposite signs.

E4.7 Find the volume of the parallelepiped whose sides are the vec-tors ~u = (1, 2, 3), ~v = (2, −2, 4), and ~w = (2, 2, 5).

4.5 MATRIX INVERSE 169

Links

[ More information about determinants from Wikipedia ] http://en.wikipedia.org/wiki/Determinant

http://en.wikipedia.org/wiki/Minor_(linear_algebra)

4.5 Matrix inverse

In this section, we’ll learn four different approaches for computing the inverse of a matrix. Since knowing how to compute matrix inverses is a pretty useful skill, learning several approaches is hardly overkill.

Note that the matrix inverse is unique, so no matter which method you use to find the inverse, you’ll always obtain the same answer.

You can verify your calculations by computing the inverse in different ways and checking that the answers agree.