Math 307Spring, 2003

Hentzel

Time: 1:10-2:00 MWFRoom: 1324 Howe Hall

Instructor: Irvin Roy HentzelOffice 432 Carver

Phone 515-294-8141E-mail: hentzel@iastate.edu

http://www.math.iastate.edu/hentzel/class.307.ICN

Text: Linear Algebra With Applications, Second Edition Otto Bretscher

Wednesday, April 2 Chapter 6.1 Page 250 Problems 8,17,44,48Main Idea: There is a magical number associated to a matrix called the determinant. Key Words: Determinants, Elementary Row OperationsDet[A]=SUM sgn(p) a1p(1) a2p(2) ... a np(n) all p Goal: Learn the definition of a determinant. how the elementary row operations affect the determinant. det (A B) = det (A) det (B)

Today we do the following.

1. Give definition of the determinant of A n x n

2. Show how the elementary row operations affect the determinant.

3. Show that det [ A B] = det [A] det[ B].

The definition in English. Given A n x n

(a) Pick n elements of A no two of which are in the same row or column.

(b) Multiply then together.(c) Calculate the appropriate sign. (d) Do this for all possible choices of n

elements. (e) The sum of these terms is the determinant

of A.

The definition using summation notation.Det[A] = SUM sgn(p) a1p(1) a2p(2) ... a np(n)

all p The n elements are from n different rows, the one in the first row is a 1 p(1) ; the one in the second row is a 2 p(2) ; etc. Sgn(p) is the calculated sign. Sgn(p) depends on the second coordinates of the chosen elements.

The sign of the term is computed by counting the number of times that a larger number precedes a smaller number in the secondsubscripts p(1) p(2) ... p(n).

If that number is even the sgn is +1, if that number is odd, then the sgn is -1.

Example: a14 a23 a32 a41

look at 4 3 2 1 there are six instances where a larger precedes a smaller. The sgn is +1.

Example: a11 a23 a32 a44

look at 1 3 2 4 there is one instance where a larger number precedes a smaller. The sgn is -1.

Example: | a11 a12 a13 | | a21 a22 a23 | | a31 a32 a33 |

+a11 a22 a33 -a13 a22 a31

+a12 a23 a31 -a11 a23 a32

+a13 a21 a32 -a12 a21 a33

Now we look at how the elementary row operations affect the determinant.

(1) Switching two rows changes the sign of the determinant.

(2) Multiply a row by a scalar c multiplies the determinant by c.

(3) Adding a multiple of one row to another does not change the determinant.

Elementary Row Operation (1).We first show that the statement is true if we switch two adjacent rows. When the rows are switched, the sign is computed with the positions p(i) and p(i+1) interchanged. This interchange increases the number of times a larger precedes a smaller when p( i ) < p(i+1).This interchange decreases the number of times a larger precedes a smaller When p( i ) > p(i+1). Since the sign changes for each term, the sign of the determinant is changed.

If the rows switched are not adjacent, we do a series of adjacent switches which have the end result we want. Ri ------- xx | xx | r rows xx | ------| RjWe switch the Ri with each of the intermediate rows,Switch Ri and Rj, and then switch Rj with the intermediate rows. The final result has only Ri and Rj switched. In the process we made 2r+1 switches. The determinant has changed sign 2r+1 times and now is opposite what it originally was.

Elementary Row Operation (2) Multiplying a row by a scalar c multiplies the determinant by c. | ---R1 --- | | --- R1 ----| | ---R2 --- | | --- R2 ----| | --- --- | | --- ----| Det | c Ri | = c Det | Ri | | --- --- | | --- ----| | --- --- | | --- ----| | ---Rn --- | | --- Rn ----|

If we multiply the elements of a row by c, there appearsexactly one c in each of the terms. We can factor the c out and place it before the summation.

If A’ is A with row i multiplied by c then

Det[ A’ ] =SUM sgn(p) a 1 p(1) ... (c a i p(i) ).... a n p(n).

all p = c SUM sgn(p) a 1 p(1) ... a i p(i) .... a n p(n).

all p = c Det[ A ].

Elementary Row Operation (3). Adding a multiple of one row to another does not change the determinant. | --- R1 --- | | --- R1 ----| | --- R1 ----| | Ri + c Rj | | --- Ri ----| | --c Rj ----| | --- --- | | --- ----| | --- ----| Det | --- --- | = | --- ----| + | --- ----| | --- Rj --- | | --- Rj ----| | --- Rj ----| | --- --- | | --- ----| | --- ----| | --- Rn --- | | --- Rn ----| | --- Rn ----|

This is true because each term in the summation has one entry from Ri + c Rj.

The term can be expanded into two pieces. Put the part from Ri into the first and the part from c Rj into the second.

| --- R1 ----| | --- R1 ----| | --- Ri ----| | -- Rj ----| | --- ----| | --- ----| = | --- ----| +c | --- ----| | --- Rj ----| | --- Rj ----| | --- ----| | --- ----| | --- Rn ----| | --- Rn ----|

The c can be brought outside by using Elementary Row Operation (2)

| --- R1 ----| | --- Ri ----| | --- ----| = | --- ----| + 0 | --- Rj ----| | --- ----| | --- Rn ----|

The determinant of a matrix with two identical rows has to be zero.

If we switch the identical rows, the sign of the determinant must change by Elementary Row Operation (1).

But since the rows were identical, the determinant has to be the same.

Det(A) = - Det(A) only happens when Det(A) = 0.

We can summarize the above results in this way.

If E is an Elementary Row Operation Matrix, then

Det [ E A] = Det [ E ] Det[ A ] for any A n x n.

Theorem: A is invertible <==> Det[A] = 0.

Proof: A is invertible <==> RCF(A) = I.

If RCF(A) = I, then Det[ RCF(A) ] = 1.

If RCF(A) =/= I, then Det[ RCF(A) ] = 0 since RCF(A) has a row of zeros.

Thus A is invertible <==> Det[ RCF(A) ] =/= 0.

Reduce A to row canonical form using elementary row operations. Ek Ek-1 ... E3 E2 E1 A = RCF(A).

Using the fact that Det[ E A ] = Det[ E] Det[ A ]

Det [Ek] … Det [E1] Det [A] = Det[ RCF(A) ].

Since each of the factors Det[ Ei ] is nonzero,

Det[A] = 0 <==> Det[ RCF( A ) ] = 0.

Combine this with the first statement says that

A is invertible <==> Det [A] =/= 0.

Theorem: Det [ A B ] = Det [A] Det [B].

Proof. If A is not invertible, then A B is not invertible and so Det[A] = 0, Det[A B] = 0

Det [A B ] = Det [A] Det [B].

If A is invertible, then A is a product of Elementary Row Operation Matrices.

A = E1 E2 ... Ek.

Det[ A B ] = Det[ E1 E2 ... Ek B]

= Det [E1] Det [E2] ... Det [Ek] Det [B]

= Det [E1 E2 ... Ek] Det [B]

= Det [A] Det [B].