Linear Algebra Homework 5: Determinants and Inverse Matrices | Assignments Linear Algebra

MATH 332 - Linear Algebra July 2, 2009

Homework 5, Summer 2009 Due: July 6, 2009

Determinants - Row-Reductions - Properties - Inverse Matrices - Volumes

1. Given the following for matrices:

A="a b

c d #,B="c d

a b #,C="a b

kc kd #,D="a+kc b +k d

c d #.

Calculate the determinants of the previous matrices by theorem 2.2.4. In each case, state the row-operation on Aand

describe how it effects the determinant.

2. The following questions illustrate some important properties of the determinant.

(a) The determinant is not, in general, a linear mapping. That is, det: Rn×n→Ris not, in general, such that,

det(A+B) = det(A)+det(B). The determinant is, in general, multilinear.1Show this for the domain R3×3by verifying

that det(A) = det(B) + det(C), where A,B,Care given as,

A=





a11 a12 u1+v1

a21 a22 u2+v2

a31 a32 u3+v3





,B=





a11 a12 u1

a21 a22 u2

a31 a32 u3





,C=





a11 a12 v1

a21 a22 v2

a31 a32 v3





.

(b) Show that if Ais invertible, then det(A−1) = 1

det(A).

(d) Let Ube a square matrix such that UtU=I. Show that det(U) = ±1.

(e) Find a formula for det(rA) where A∈Rn×nand r∈R.

3. Given,

A=





1 5 −3

3−3 3

2 13 −7





.(1)

(a) Find det(A) using cofactor expansion.

(b) Find det(A) using row reduction to echelon form.

4. Given,

A=





367

021

234





.

Calculate the adjugate of Aand using theorem 3.3.8 calculate A−1.

5. The determinant has a geometric interpretation. In R2, det(A) is the area of the parallelogram formed by the two vectors

a1,a2, where A= [a1a2] . In R3, det(A) is the volume of the parallelepiped formed by the three vectors a1,a2,a3, where

A= [a1a2a3].

Using the concept of volume, explain why the determinant of a 3 ×3 matrix Ais zero if and only if Ais not invertable.

Note: No credit will be given for the use of Theorem 3.2.4. Also, note that there are two proofs here. The forward proof

should assume that det(A)= 0 and conclude that Ais singular by using the geometry formed by the column vectors. The

backward proof should start assuming Ais singular and conclude that the parallelepiped volume is zero. The two proofs

together prove the if and only if statement above.

Hint: Use the invertible matrix theorem of 2.3 and a geometric description of linearly dependent vectors in R3.

1A multilinear map is a mathematical function of several vector variables that is linear in each variable. That is, if all columns except one are fixed,

then the determinant is a linear function of that one column. See http://en.wikipedia.org/wiki/Multilinear_map for more information.

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MATH 332 - Linear Algebra July 2, 2009 Homework 5, Summer 2009 Due: July 6, 2009 Determinants - Row-Reductions - Properties - Inverse Matrices - Volumes

Given the following for matrices:

A =

[

a b c d

]

, B =

[

c d a b

]

, C =

[

a b kc kd

]

, D =

[

a + kc b + kd c d

]

Calculate the determinants of the previous matrices by theorem 2.2.4. In each case, state the row-operation on A and describe how it effects the determinant.

The following questions illustrate some important properties of the determinant. (a) The determinant is not, in general, a linear mapping. That is, det: Rn×n^ → R is not, in general, such that, det(A+B) = det(A)+det(B). The determinant is, in general, multilinear.^1 Show this for the domain R^3 ×^3 by verifying that det(A) = det(B) + det(C), where A, B, C are given as,

A =

a 11 a 12 u 1 + v 1 a 21 a 22 u 2 + v 2 a 31 a 32 u 3 + v 3

 , B =

a 11 a 12 u 1 a 21 a 22 u 2 a 31 a 32 u 3

 , C =

a 11 a 12 v 1 a 21 a 22 v 2 a 31 a 32 v 3

(b) Show that if A is invertible, then det(A−^1 ) = (^) det^1 (A). (c) Let A and P be square matrices such that P−^1 exists. Show that det(PAP−^1 ) = det(A). (d) Let U be a square matrix such that UtU = I. Show that det(U) = ±1. (e) Find a formula for det(rA) where A ∈ Rn × n and r ∈ R.

Given,

A =

(a) Find det(A) using cofactor expansion. (b) Find det(A) using row reduction to echelon form.

Given,

A =

Calculate the adjugate of A and using theorem 3.3.8 calculate A−^1.

The determinant has a geometric interpretation. In R^2 , det(A) is the area of the parallelogram formed by the two vectors a 1 , a 2 , where A = [a 1 a 2 ]. In R^3 , det(A) is the volume of the parallelepiped formed by the three vectors a 1 , a 2 , a 3 , where A = [a 1 a 2 a 3 ].

Using the concept of volume, explain why the determinant of a 3 × 3 matrix A is zero if and only if A is not invertable. Note: No credit will be given for the use of Theorem 3.2.4. Also, note that there are two proofs here. The forward proof should assume that det(A)= 0 and conclude that A is singular by using the geometry formed by the column vectors. The backward proof should start assuming A is singular and conclude that the parallelepiped volume is zero. The two proofs together prove the if and only if statement above. Hint: Use the invertible matrix theorem of 2.3 and a geometric description of linearly dependent vectors in R^3. (^1) A multilinear map is a mathematical function of several vector variables that is linear in each variable. That is, if all columns except one are fixed, then the determinant is a linear function of that one column. See http://en.wikipedia.org/wiki/Multilinear_map for more information.

Linear Algebra Homework 5: Determinants and Inverse Matrices, Assignments of Linear Algebra

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