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Lesson Worksheet: Inverse of a Matrix: The Adjoint Method Mathematics • 10th Grade

In this worksheet, we will practice finding the inverse of 3 × 3 matrices using the adjoint method.

Q1:

Find the cofactor matrix of 𝐴=7βˆ’5βˆ’8βˆ’3βˆ’7βˆ’20βˆ’4βˆ’8.

  • A48βˆ’8βˆ’46βˆ’24βˆ’56381228βˆ’64
  • B75βˆ’83βˆ’7204βˆ’8
  • C48βˆ’2412βˆ’8βˆ’5628βˆ’4638βˆ’64
  • D4824128βˆ’56βˆ’28βˆ’46βˆ’38βˆ’64

Q2:

Find the adjoint matrix of the matrix 𝐴=2βˆ’72βˆ’9βˆ’7βˆ’9βˆ’85βˆ’4.

  • A73βˆ’18773680βˆ’10146βˆ’77
  • B731877βˆ’3680βˆ’101βˆ’46βˆ’77
  • C7336βˆ’101βˆ’18846770βˆ’77
  • D2729βˆ’79βˆ’8βˆ’5βˆ’4

Q3:

Find the multiplicative inverse of the matrix ο˜βˆ’5000βˆ’5000βˆ’5.

  • Aβˆ’1125250002500025
  • B250002500025
  • Cβˆ’1125ο˜βˆ’25000βˆ’25000βˆ’25
  • Dο˜βˆ’25000βˆ’25000βˆ’25

Q4:

True or False: If 𝐴 is any given square matrix of order 𝑛, then 𝐴×(𝐴)=(𝐴)×𝐴=|𝐴|𝐼adjadj, where 𝐼 is the identity matrix of order 𝑛.

  • AFalse
  • BTrue

Q5:

Consider the matrix 𝐴=2140530010. Find its inverse, given that it has the form 𝐴=ο™π‘‹π‘π‘ž0π‘Œπ‘Ÿ00𝑍ο₯, where 𝑋, π‘Œ, 𝑍, 𝑝, π‘ž, and π‘Ÿ are numbers you should find.

  • A𝐴=⎑⎒⎒⎒⎒⎒⎣14βˆ’1120161300110⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • B𝐴=⎑⎒⎒⎒⎒⎒⎣141120131500110⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • C𝐴=⎑⎒⎒⎒⎒⎒⎣121140151300110⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • D𝐴=⎑⎒⎒⎒⎒⎒⎣12βˆ’110βˆ’17100015βˆ’35000110⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • E𝐴=⎑⎒⎒⎒⎒⎒⎣12βˆ’1130161300110⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦

Q6:

Consider the matrix 103101310. Determine whether the matrix has an inverse by finding whether the determinant is nonzero. If the determinant is nonzero, find the inverse using the formula for the inverse which involves the cofactor matrix.

  • AThere is no inverse because its determinant equals zero.
  • BIt has an inverse, which is βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’12321232βˆ’92βˆ’12010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.
  • CIt has an inverse, which is βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1232032βˆ’92112βˆ’120⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.
  • DIt has an inverse, which is ο˜βˆ’1313βˆ’9βˆ’1020.
  • EIt has an inverse, which is ο˜βˆ’1303βˆ’921βˆ’10.

Q7:

Suppose that 𝐴𝑋=𝐡, where 𝐴=4βˆ’135043βˆ’32,𝐡=2103βˆ’4527βˆ’6,

and 𝑋 is a 3Γ—3 matrix.

Calculate the inverse of 𝐴 and use it to find 𝑋.

  • Aο˜βˆ’512βˆ’11βˆ’1βˆ’117βˆ’1615
  • Bο˜βˆ’22βˆ’3βˆ’2βˆ’41βˆ’110βˆ’8
  • C1129βˆ’231833βˆ’24129βˆ’27
  • D11βˆ’29βˆ’23βˆ’1833241βˆ’29βˆ’27
  • Eο˜βˆ’5βˆ’12βˆ’111βˆ’1βˆ’171615

Q8:

Determine whether the matrix 123021267 has an inverse by finding whether the determinant is nonzero. If the determinant is nonzero, find the inverse using the inverse formula involving the cofactor matrix.

  • AThere is no inverse because the determinant equals zero.
  • B8βˆ’2βˆ’4βˆ’412βˆ’412
  • C82βˆ’441βˆ’2βˆ’4βˆ’12
  • D8βˆ’4βˆ’4βˆ’211βˆ’422
  • E84βˆ’421βˆ’1βˆ’4βˆ’22

Q9:

Consider the matrix 103101310. Determine whether the matrix has an inverse by finding whether the determinant is nonzero. If the determinant is nonzero, find the inverse using the formula for the inverse which involves the cofactor matrix.

  • AIt has an inverse, which is βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’12321232βˆ’92βˆ’12010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.
  • BIt has an inverse, which is ο˜βˆ’1303βˆ’921βˆ’10.
  • CIt has an inverse, which is ο˜βˆ’1313βˆ’9βˆ’1020.
  • DIt has an inverse, which is βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1232032βˆ’92112βˆ’120⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.
  • EThere is no inverse because its determinant equals zero.

Q10:

Find the inverse of the matrix οšπ‘’π‘‘π‘‘π‘’βˆ’π‘‘π‘‘π‘’βˆ’π‘‘βˆ’π‘‘ο¦.cossinsincoscossin

  • A𝑒0𝑒(𝑑+𝑑)βˆ’2𝑑(π‘‘βˆ’π‘‘)(π‘‘βˆ’π‘‘)2π‘‘βˆ’(𝑑+𝑑)sincossinsincossincoscossincos
  • B⎑⎒⎒⎒⎒⎒⎣12𝑒012𝑒12(𝑑+𝑑)βˆ’π‘‘π‘‘12(π‘‘βˆ’π‘‘)π‘‘βˆ’12(𝑑+𝑑)⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦sincossincossincoscossincos
  • C⎑⎒⎒⎒⎒⎒⎣12𝑒012𝑒12(𝑑+𝑑)βˆ’π‘‘12(π‘‘βˆ’π‘‘)12(π‘‘βˆ’π‘‘)π‘‘βˆ’12(𝑑+𝑑)⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦sincossinsincossincoscossincos
  • D⎑⎒⎒⎒⎒⎣12𝑒12(𝑑+𝑑)12(π‘‘βˆ’π‘‘)0βˆ’π‘‘π‘‘12𝑒12(π‘‘βˆ’π‘‘)βˆ’12(𝑑+𝑑)⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦sincossincossincossincossincos
  • E𝑒(𝑑+𝑑)(π‘‘βˆ’π‘‘)0βˆ’2𝑑2𝑑𝑒(π‘‘βˆ’π‘‘)βˆ’(𝑑+𝑑)sincossincossincossincossincos

This lesson includes 42 additional questions and 117 additional question variations for subscribers.

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