Linalg_Determinants.html

Properties of Determinants

Linear Algebra Project

Objectives:

Use Maple

1. To create random square matrices.

2. To study properties of determinants.

Solved Problem:

1. Construct a square matrix, find its transpose and verify that determinants of both the matrices are equal.

2. Find a relation between the inverse of a square matrix with the determinant of the matrix.

3. Study the effect of the following on a determinant and write a conclusion:

a) Multiply a row of the determinant by a constant.

b) Interchange two rows of a determinant.

c) Construct a determinant with a row consisting of all zeros.

d) Construct a determinant with two identical rows.

e) Add a multiple of one row to another.

Solution:

1. A random square matrix is created by randmatrix operation.

>

with(linalg):

>

A := randmatrix(3,3);

This matrix changes every time you press <enter> after the above command in the downloaded version in Maple. The following discussion is based on the above matrix.

>

ATr := transpose(A);

>

det(A);

>

det(ATr);

This shows that the determinants of a matrix and its transpose have the same value.

An important consequence is that the properties for the rows of a determinant apply to columns by merely substituting the word column/s for row/s in the properties.

2. In this part we show that the inverse of a matrix is the adjoint of the matrix divided by its determinant. The adjoint A is formed by taking the values of the minors of the entries along with the sign for that entry.

>

inverse(A);

>

adjoint(A);

>

adjoint(A)/det(A);

From the above we see that the inverse(A) = adjoint(A)/det(A).

3. a) We multiply the first row of the matrix A by c and find the determinant of the resulting matrix. We copy and paste the matrix A from the Maple output in part 1, multiply by the first row by 3 and call the matrix Ac.

>

A := matrix([[63, 57, -59], [45, -8, -93], [92, 43, -62]]);

>

Ac:= matrix( [[189, 171, -177], [45, -8, -93], [92, 43, -62]]);

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det(Ac);

>

3*det(A);

Conclusion: If a row of a determinant is multiplied by a constant, the value of the determinant is multiplied by the constant.

b) We copy and paste A, and interchange the first and second rows and the value of the determinants.

>

A21 := matrix([ [45, -8, -93],[63, 57, -59], [92, 43, -62]]);

>

det(A21);

Conclusion: If two rows are interchanged the sign of the value of the determinant is changed.

c) We consider the matrix A with last row substituted by zeros and call the matrix B.

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B := matrix([[63, 57, -59], [45, -8, -93], [0, 0, 0]]);

>

det(B);

Conclusion: If a row of a determinant is zero the value of the determinant is zero.

d) We take the matrix A and make the first rows equal and call the matrix C.

>

C := matrix([[-85, -55, -37], [-85, -55, -37], [79, 56, 49]]);

>

det(C);

Conclusion: If the row of a determinant consists of two identical rows, the value of the determinant is zero.

e) We multiply 3 times of first row of A with the second row, call the resulting matrix E and compute the determinant of E.

>

E := matrix([[63, 57, -59], 3*[63, 57, -59]+[45, -8, -93], [92, 43, -62]]);

>

det(E);

Conclusion: If a multiple of a row is added to another row, the value of the new determinant equals the value of the original determinant.

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ASSIGNMENT

Problem 1:

Use the matrix A =

a) Find the transpose of A. What is the relation between the determinants of A and transpose of A?

b) Find the inverse of A, adjoint of A and determinant of A and show that

inverse(A) = adjoint(A)/det(A).

c) Describe the effect on the determinant of A

i) if a column of A is multiplied by a constant.

ii) if two columns are interchanged.

iii) if one column consists of all zeros.

iv) if two columns are identical.

v) if a multiple of a column is added to a column of A.

Problem 2:

Generate a random 4 x 4 matrix and answer all the questions posed in Problem 1.

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MSIP Grant #P120A80089-98: "Three Urban Calculus Reform programs: Adopting the Best" 1998-2001, MSEIP Grant #P120A010031: "Four Colleges: Calculus + Enhancements" 2001-04