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Mathematics 1229A/B Lecture Notes - Lecture 23: Asana, Axa

31 views6 pages

22 Dec 2019

School

Western University

Department

Course

Mathematics 1229A/B

Professor

Mohsen Mollahajiaghaei

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Propertiesof Determinants

Theorem

letAand Abe

square

matricies

that

are

thethesameexceptthat

inonerow or in one

column ofA

each

entry of AismultipliedbyascalarcThende

ta ec

de

1A Thatis ccanbe

factoredout ofthe

row oncolumn

Example

letAIg weknow

that detail

del

B3

de

1A

LetBµkg 32 FinddetB 347

21

Example

LetALac

bd bea22matrix

suchthat de

1A 4Findthedeterminantof BLac

bd

del

B2

de

1A

a2C4

8

Example

Let ALacbd bea22matrix

such

thatdetA 4Find

thedeterminantofB26

del

B3

detµd3det 22 bd

32det Ibd

6L4

24

Example

letAgadabe bea33matrix

suchthatdetA 1Find

thedeterminant of

I

det

B2

de

1gd

Eben

Egf

212 det Idg

g

2C

22det god

i8

Cl

8

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Theorem

let AbethesquarematrixofordernThen

DetCA ch

de

1A

Example delCAC1de

1A

Example

betAgFinddeta

i2io

4a22

de

1Aa2det

µIfII

SuZdag

2

Co

o

Theorem

if asquare

matrixAhasarowthatis ascalar

multipleofanotherrow orcolumnthat is a

scalar

multipleof

anothercolumn

1thende1A0

Example

letAIda

abe 2and BIg Idg ThendelAoanddel

Bo

Example

aµ22,1 Findde

1A

c2e de

1Ao

Example

LetAIg 2Find

de

1A

deta ade

1A aade

1Aaa3

det

Aand

et

Acy

Idef gIz

Is ooo

Rs 7RThende

1A o

Example

LetAI4and BIFindde

1AandDet

B

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MATH 1229A/B Full Course Notes

MATH 1229A/B Full Course Notes

Document Summary

Theorem leta and a besquarematriciesthatarethethesameexceptthatinonerow or in onecolumn ofa eachentry of a ismultipliedbya scalarc thendeta e cde1a thatis c canbefactoredout ofthe row oncolumn. Leta lacbd bea 2 2 matrixsuchthat de1a 4 findthedeterminantof b lacbd delb 2de1a a2c4. Let a lacbd bea 2 2 matrixsuchthatdeta 4 findthedeterminantofb 26. Example let a gadabe be a 3 3matrixsuchthatdeta 1 findthedeterminant of. 4 a 2 2 de1aa 2det i f i i. Theorem if asquarematrixahas a rowthatis ascalarmultipleofanotherrow orcolumnthat is ascalarmultipleof anothercolumn1thende1a 0. 2 and b ig idg thendela o anddelb o a 22,1 findde1a c 2e de1a o. Findde1a deta a de1a aade1a a a3deta andetacy. 4 l use4 casca to de1b andetb acadetbat abdetbs. Example let a idg andde1a 4 finddetbwhere b a ebnadg de1be de1a. Ra ry deta cn de1 gig ig deta i b 5c2 so. Rowoperations sofar wehaveseenhowdeterminantsofsquarematriciesbehavewithrespecttotwoofthethreeelementaryrow operations if a rowis multipliedbyascalarthedeterminantismultipliedbythesamescalar iftworowsareinterchangedthesignofthedeterminant is reversed. Theorem if a scalarmultipleofonerowcorcolumn isaddedtoanotherrowcorcolumn of a squarematrix thedeterminantis unchanged.

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Related Questions

j

PLEASE NEED HELP WITH THIS TEST!!!

Use the figure below for Exercises 32-34. Find m 2 if mST = 132. Find m 4 if mRQT = 136. Find mQTS if m 1 = 27 and m 2 = 60. Classify the triangle as right, acute, or obtuse if the measures of its sides are 31, 45, and 56. Find the center and radius of the circle with equation (x + 6)2 + (x - 5)2 = 64. Find the lengths of MW- and MX-. Round your answers to the nearest tenth. Find the area of the trapezoid. Find the value of x if BD = 8 and BC = 17. Find the area of an equilateral triangle with perimeter of 60 in. Round your answer to the nearest hundredth. Find the values of x and y. Leave your answers in simplest radical form. The lengths of the sides of a right triangle are 18 ft. 24 ft. and 30 ft. What is the area of the triangle? 216 ft2 270 ft2 432 ft2 cannot be determined Find the volume of the cylinder in terms of pi. Find the value of x. Determine which of the following will tessellate. Decagon pentagon octagon triangle Determine the angle of rotation of Figure 1 to Figure 2. Figure 1 Figure 2 Given CKMS ~ DLNT. If the area of CKMS is 32 cm2, what is the area of DLNT? Find the height of a regular square prism with base edge of 6 ft and volume of 504 ft 3. What is the distance between (-12.6) and (8, -9)? Find the value of x. Assume that segments that appear to be tangent are tangent. 40 50 60 100 Find the values of x and y. If Delta DHT is similar to a triangle whose sides have lengths 4, 7, and 9. which of the following could be the perimeter of Delta DHT? 20 40 60 any of these Identify the hypothesis and conclusion of the conditional. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Which of the following is the converse of the conditional in Exercise 3? If a point is congruent from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If a point is equidistant from a segment, then it is on a segment perpendicular to the bisector. If a point is not on the perpendicular bisector of a segment, then it is not equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. What is the equation of a line perpendicular to that contains (4, -3)? 2x - y = 11 x - 2y = 10 2x - y = 5 x - 2y = -2 Find the length of BD. YW rightarrow bisects . Find m . Given Delta JKL ~ Delta MNP with a scale factor of 3 : 4. If Delta JKL has perimeter 57, what is the perimeter of Delta MNP? If Delta KLM Delta BCS, then which of the following is not true? Use the figure below for Exercises 10-13. Identify the pairs of angles as: alternate interior angles corresponding angles same-side interior angles vertical angles A circle has radius 17 m. Find its area and circumference to the nearest tenth. Find the coordinates of the midpoint of (9, 1) and (3, -5). Write the equation of a line in slope-intercept form that is parallel to y = - x + 8 and contains (-3 ,-5). For each pair of triangles, state the postulate or theorem you can use to prove the triangles congruent. If the triangles cannot he proven congruent, write not possible.

Show transcribed image text Use the figure below for Exercises 32-34. Find m 2 if mST = 132. Find m 4 if mRQT = 136. Find mQTS if m 1 = 27 and m 2 = 60. Classify the triangle as right, acute, or obtuse if the measures of its sides are 31, 45, and 56. Find the center and radius of the circle with equation (x + 6)2 + (x - 5)2 = 64. Find the lengths of MW- and MX-. Round your answers to the nearest tenth. Find the area of the trapezoid. Find the value of x if BD = 8 and BC = 17. Find the area of an equilateral triangle with perimeter of 60 in. Round your answer to the nearest hundredth. Find the values of x and y. Leave your answers in simplest radical form. The lengths of the sides of a right triangle are 18 ft. 24 ft. and 30 ft. What is the area of the triangle? 216 ft2 270 ft2 432 ft2 cannot be determined Find the volume of the cylinder in terms of pi. Find the value of x. Determine which of the following will tessellate. Decagon pentagon octagon triangle Determine the angle of rotation of Figure 1 to Figure 2. Figure 1 Figure 2 Given CKMS ~ DLNT. If the area of CKMS is 32 cm2, what is the area of DLNT? Find the height of a regular square prism with base edge of 6 ft and volume of 504 ft 3. What is the distance between (-12.6) and (8, -9)? Find the value of x. Assume that segments that appear to be tangent are tangent. 40 50 60 100 Find the values of x and y. If Delta DHT is similar to a triangle whose sides have lengths 4, 7, and 9. which of the following could be the perimeter of Delta DHT? 20 40 60 any of these Identify the hypothesis and conclusion of the conditional. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Which of the following is the converse of the conditional in Exercise 3? If a point is congruent from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If a point is equidistant from a segment, then it is on a segment perpendicular to the bisector. If a point is not on the perpendicular bisector of a segment, then it is not equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. What is the equation of a line perpendicular to that contains (4, -3)? 2x - y = 11 x - 2y = 10 2x - y = 5 x - 2y = -2 Find the length of BD. YW rightarrow bisects . Find m . Given Delta JKL ~ Delta MNP with a scale factor of 3 : 4. If Delta JKL has perimeter 57, what is the perimeter of Delta MNP? If Delta KLM Delta BCS, then which of the following is not true? Use the figure below for Exercises 10-13. Identify the pairs of angles as: alternate interior angles corresponding angles same-side interior angles vertical angles A circle has radius 17 m. Find its area and circumference to the nearest tenth. Find the coordinates of the midpoint of (9, 1) and (3, -5). Write the equation of a line in slope-intercept form that is parallel to y = - x + 8 and contains (-3 ,-5). For each pair of triangles, state the postulate or theorem you can use to prove the triangles congruent. If the triangles cannot he proven congruent, write not possible.

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