MATA33 Determinant notes, mostly stolen from Prof Eric Moore

Definition: determinant (i) The determinant of a 1 by 1 matrix [a] is a. (ii) Suppose a definition is provided for a n−1 by n−1 determinant. Define the determinant of the n*n matrix by:

=∑ 1 det

where is the matrix obtained from A by deleting the ith

row and jth

column. The matrix is sometimes called the

ijth

minor matrix of A. Think of this approach as the definition by expansion along the first row or expansion by minors along the first row.

nnnn

n

n

aaa

aaa

aaa

...

......

......

......

...

...

det

21

21221

11211

Properties of Determinants Let A be a n x n matrix. If A has a row or column of zeros, det A = 0. If A is a triangular matrix, det A is the product of the elements on the main diagonal. det I = 1.

det A = det AT

.

Interchanging any two rows (or columns) multiplies the

determinant by −1.

Properties of Determinants

Since det A = det AT

let’s state the remaining properties only

in terms of rows.

If a row of A is multiplied by a constant k, the

determinant of the resulting matrix is k det A.

o Note that det(kA)= kn det A.

Let A, B, C be n × n matrices that are identical except

that the ith

row of A is the sum of the ith

rows of B and C

then det A = det B + det C.

o but det (E+F) ≠ det E + det F in general.

o This property + previous property =⇒

determinant is linear in each row.

If two distinct rows are identical then det A = 0.

If two distinct rows are proportional then det A = 0. .

Adding a multiple of one row to another row does not

change the value of the determinant.

det(AB) = (det A) (det B).

The determinant of a matrix consists of sums and

products of its entries. If the entries are polynomials in

some variable, say x, then the determinant is a polynomial

in x. Often it is of interest to know the values of x that

make the determinant zero.

A square matrix is invertible if and only if det A ≠ 0.

The homogeneous linear system AX = 0 has the unique

solution X = 0 if and only if det A ≠ 0.

Let be an n × n matrix. We define the cofactor of aij denoted cij by

1

The n × n matrix C = [cij] is called the cofactor matrix of A.

(Recall that is the matrix obtained from A by deleting

the ith

row and jth

column and is sometimes called the ijth

minor

matrix of A. det can be called the ijth

minor of A or the

minor of the element aij of A)

In this context, a cofactor is sometimes called a signed minor.

Note that cij is a scalar (real number) but ij is an

(n − 1) × (n − 1) matrix.

We can now restate the definition of determinant in terms of cofactors. (i) The determinant of a 1 by 1 matrix [a] is a.

(ii) Suppose a definition is provided for a n−1 by n−1 determinant. Define = ∑ = a11c11 + a12c12 + …….. + a1nc1n

where cij is the cofactor of aij.

nnnn

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aaa

aaa

...

......

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......

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det

21

21221

11211

Cofactor Method of Inverting a Matrix The transpose of the cofactor matrix C of A is called the

classical adjoint of A and denoted by adj A; i.e., adj A = CT

. Theorem: If A is any square matrix, then A ( adj A ) = (det A ) I = ( adj A ) A. In particular, if det A ≠ 0, the inverse of A is given by

,

where adj A = CT

, C being the cofactor matrix of A. (Comment: this is the classical mathematician’s way of talking about matrix inverses, and is good for discussing properties of inverses. Faster methods including reduction are used more in practice.)

Example on Matrix Inversion by Cofactors

3 52 7

1

: 7 25 3

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7 5

2 321 10

7/11 5/112/11 3/11

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MATA33 Final Summer 2009 MCQ8

Assume A is an n x n matrix where n ≥ 2: Exactly how many of the following mathematical statements are always true ? (i) det(2A) = 2n det(A) (ii) det(In + A) = 1 + det(A) (iii) If AX = 0 has infinitely many solutions, then A is not invertible. (iv) If AB = AC, then B = C (v) If R is the reduced form of A and R has exactly n non-zero rows, then A is invertible. (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (B) Solution (i) No: every element is *2 so det(2A)=2n det(A) (ii) No – lots of counterexamples (iii) Yes – if A-1 exists then A-1 multiplies zero vector to get the unique trivial solution, all zeroes (iv) No – eg A is the zero matrix (v) Yes – there’s n unknowns, n lin indept eqns

MA

ATA33 W099 T2 LA

AQ1

MMATA3

33 W09 T2 LLAQ1 ((cont)

MATA33 W09 T2 LAQ1 (Cont)

Alternative derivation of M-1, using for x=6 that detM=2(6-7)2=2.

2 6 62 7 62 7 7

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2

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Which checks with Prof Grinnell’s answer

MMATA333 W099 T2 LLAQ6 ((Cont)

MAT

TA33 S09 T22 MCQQ1

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MATA33 S09 T2 LAQ3b (Alternative solution)

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Which checks with Prof Grinnell’s answer by reduction

MAT

TA33 W

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Cramer’s Rule

If A is an invertible n × n matrix, the solution to the system

AX = B

of n equations in the variables x1,x2, ...., xn is given by:

,

,

,

where, for each k, Ak is the matrix obtained from A by

replacing the kth.

column of A by B.

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3x + 2y = 27 x - y = -1 Cramer says:

,

x=

5

y=

6