Chapter 2B Determinants

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Chapter 2B Determinants
When we look at a particular square matrix, the
question of whether it is nonsingular is one of
the first things that we ask. This chapter
develops a formula to determine this.

• 2B.1 The Determinant and Evaluation of a Matrix
• 2B.2 Properties of Determinants
• 2B.3 Eigenvalues and Application of Determinants
• 2B.4 Geometry of Determinants Determinants as
  Size Functions

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2B.1 The Determinant of a Matrix

• The determinant of a 2 2 matrix

• Note

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• Minor of the entry

• Cofactor of

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• Ex

• Notes Sign pattern for cofactors

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• Thm 3B.1 (Expansion by cofactors)

Let A is a square matrix of order n, then the
determinant of A is given by
(Cofactor expansion along the i-th row, i1, 2,,
n )
or
(Cofactor expansion along the j-th column, j1,
2,, n )
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• Ex The determinant of a matrix of order 3

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• Ex 5 (The determinant of a matrix of order 3)

Sol
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• The determinant of a matrix of order 3

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• Upper triangular matrix

All the entries below the main diagonal are
zeros.

• Lower triangular matrix

All the entries above the main diagonal are
zeros.

• Diagonal matrix

All the entries above and below the main
diagonal are zeros.
diagonal
upper triangular
lower triangular
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• Theorem 2B.2 Determinant of a Triangular Matrix

If A is an nxn triangular matrix (upper
triangular, lower triangular, or diagonal), then
its determinant is the product of the entries on
the main diagonal. That is
At this moment, our primary way to decide whether
a matrix is singular is to do Gaussian reduction
and then check whether the diagonal of resulting
echelon form matrix has any zeroes. We will look
for a family of functions with the property of
being unaffected by row operations and with the
property that a determinant of an echelon form
matrix is the product of its diagonal entries.
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Ex Find the determinants of the following
triangular matrices.
(a)
Sol
A (2)(2)(1)(3) 12
(a)
B (1)(3)(2)(4)(2) 48
(b)
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Keywords in This Section

• determinant ???
• minor ????
• cofactor ???
• expansion by cofactors ?????
• upper triangular matrix ?????
• lower triangular matrix ?????
• diagonal matrix ????

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2B.2 Evaluation of a determinant using elementary
operations

• Theorem 2B.3 Elementary row operations and
  determinants

Let A and B be square matrices,
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• Ex

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Note A row-echelon form of a square matrix
is always upper triangular.

• Ex Evaluation a determinant using elementary
  row operations

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• Theorem 2B.4 Conditions that yield a zero
  determinant

If A is a square matrix and any of the following
conditions is true, then det (A) 0.
(a) An entire row (or an entire column) consists
of zeros.
(b) Two rows (or two columns) are equal.
(c) One row (or column) is a multiple of another
row (or column).
The theorem states that a matrix with two
identical rows or two linear dependent rows has a
determinant of zero. A matrix with a zero row has
a determinant of zero. Note that a matrix is
nonsingular if and only if its determinant is
nonzero and the determinant of an echelon form
matrix is the product down its diagonal. This
theorem provides a way to compute the value of a
determinant function on a matrix Do Gaussian
reduction, keeping track of any changes of sign
caused by row swaps and any scalars that are
factored out, and then finish by multiplying down
the diagonal of the echelon form result.
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• Note

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• Ex (Evaluating a determinant)

Sol
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2B.2 Properties of Determinants

• Theorem 2B.5 Determinant of a matrix product

det (AB) det (A) det (B)

• Notes

(1) det (EA) det (E) det (A)
(2)
(3)
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• Ex (The determinant of a matrix product)

Find A, B, and AB
Sol
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• Theorem 2B.6 Determinant of a scalar multiple
  of a matrix

If A is an n n matrix and c is a scalar, then
det (cA) cn det (A)

• Ex

Find A.
Sol
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• Thm 2B.7 Determinant of an invertible matrix

A square matrix A is invertible (nonsingular) if
and only if det (A) ? 0

• Ex (Classifying square matrices as singular or
  nonsingular)

Sol
A has no inverse (it is singular).
B has inverse (it is nonsingular).
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• Thm 2B.8 Determinant of an inverse matrix

• Thm 2B.9 Determinant of a transpose

• Ex

(a)
(b)
Sol
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• Equivalent conditions for a nonsingular matrix

• If A is an n n matrix, then the following
  statements are equivalent.

(1) A is invertible.
(2) Ax b has a unique solution for every n 1
matrix b.
(3) Ax 0 has only the trivial solution of zero
column vector.
(4) A is row-equivalent to In
(5) A can be written as the product of
elementary matrices.
(6) det (A) ? 0
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• Ex Which of the following system has a unique
  solution?

(a)

Sol
This system does not have a unique solution.
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(b)

• Sol

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2B.3 Introduction to Eigenvalues

• Eigenvalue problem

If A is an n?n matrix, do there exist nonzero n?1
matrices x such that Ax is a scalar multiple of
x?
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• Ex 1 (Verifying eigenvalues and eigenvectors)

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• Question
• Given an n?n matrix A, how can you find
  the eigenvalues and
• corresponding eigenvectors?

• Note

(homogeneous system)

• Characteristic equation of A?Mn?n

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• Ex (Finding eigenvalues and eigenvectors)

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Application Example of Eigenvalue-Eigenvector
Problem

• The equations of motion for identical mass
• and spring constant can be
  described by

We can obtain
Rearrange these to put them into a neater form
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A nontrivial solution occurs when the determinant
is zero
, which yields the following solutions
(eigenvalues)
With the given eigenvalues, we can find the
corresponding eigenvectors (normal modes) to be
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2.3 Applications of Determinants

• Matrix of cofactors of A

• Adjoint matrix of A

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• Thm 2B.10 The inverse of a matrix given by its
  adjoint

If A is an n n invertible matrix, then

• Ex

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• Ex

(a) Find the adjoint of A.
(b) Use the adjoint of A to find
Sol
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cofactor matrix of A
inverse matrix of A
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• Thm 2B.11 Cramers Rule

(this system has a unique solution)
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( i.e.,
)
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• Pf

A x b,
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• Ex Use Cramers rule to solve the system of
  linear equations.

Sol
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Keywords in This Section

• matrix of cofactors ?????
• adjoint matrix ????
• Cramers rule Cramer ??

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2B.4 Geometry of Determinants Determinants as
Size Functions

• We have so far only considered whether or not a
  determinant is zero, here we shall give a meaning
  to the value of that determinant.

O
One way to compute the area that it encloses is
to draw this rectangle and subtract the area of
each subregion.
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• The properties in the definition of determinants
  make reasonable postulates for a function that
  measures the size of the region enclosed by the
  vectors in the matrix.
• See this case

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• Another property of determinants is that they are
  unaffected by pivoting. Here are before-pivoting
  and after-pivoting boxes (the scalar used is
• k 0.35).

Although the region on the right, the box formed
by and , is more slanted than
the shaded region, the two have the same base and
the same height and hence the same area.
This illustrates that
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• That is, weve got an intuitive justification to
  interpret det ( , . . . , ) as the
  size of the box formed by the vectors.

Example The volume of this parallelepiped, which
can be found by the usual formula from high
school geometry, is 12.
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• The only difference between them is in the order
  in which the vectors are taken. If we take
  first and then go to , follow the
  counterclockwise are shown, then the sign is
  positive. Following a clockwise are gives a
  negative sign. The sign returned by the size
  function reflects the orientation or sense of
  the box.

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• Volume, because it is an absolute value, does not
  depend on the order in which the vectors are
  given. The volume of the parallelepiped in the
  following example, can also be computed as the
  absolute value of this determinant.

The definition of volume gives a geometric
interpretation to something in the space, boxes
made from vectors.
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Application of the map t represented with respect
to the standard bases by
will double sizes of boxes, e.g., from this
to