Worksheet: Inverse of a Matrix: The Adjoint Method

In this worksheet, we will practice finding the inverse of 3x3 matrices using the adjoint method.

Q1:

By considering the value of the determinant, determine whether the matrix 123021310 has an inverse. If so, find the inverse by considering the matrix of cofactors.

  • AIt has no inverse.
  • BIt has an inverse, 113313413313913113613513213.
  • CIt has an inverse, 113313613313913513413113213.
  • DIt has an inverse, 12532562532592515425125225.
  • EIt has an inverse, 12532542532592512562515225.

Q2:

Find the inverse of the matrix 𝑒𝑡𝑡𝑒𝑡𝑡𝑒𝑡𝑡.cossinsincoscossin

  • A𝑒0𝑒(𝑡+𝑡)2𝑡(𝑡𝑡)(𝑡𝑡)2𝑡(𝑡+𝑡)sincossinsincossincoscossincos
  • B12𝑒012𝑒12(𝑡+𝑡)𝑡𝑡12(𝑡𝑡)𝑡12(𝑡+𝑡)sincossincossincoscossincos
  • C12𝑒012𝑒12(𝑡+𝑡)𝑡12(𝑡𝑡)12(𝑡𝑡)𝑡12(𝑡+𝑡)sincossinsincossincoscossincos
  • D12𝑒12(𝑡+𝑡)12(𝑡𝑡)0𝑡𝑡12𝑒12(𝑡𝑡)12(𝑡+𝑡)sincossincossincossincossincos
  • E𝑒(𝑡+𝑡)(𝑡𝑡)02𝑡2𝑡𝑒(𝑡𝑡)(𝑡+𝑡)sincossincossincossincossincos

Q3:

Use technology to find the inverse of the matrix 𝐴=332161312.

  • A𝐴=174134151125191215
  • B𝐴=180134151125191215
  • C𝐴=180131194121215515
  • D𝐴=180134151125191215
  • E𝐴=174134151125191215

Q4:

Use technology to find the inverse of the matrix 𝐴=224111256.

  • A𝐴=16143846220
  • B𝐴=16182442360
  • C𝐴=122182442360
  • D𝐴=16182442360
  • E𝐴=122182442360

Q5:

Use technology to find the inverse of the matrix 𝐴=110103052.

  • A𝐴=1171523223551
  • B𝐴=1131523223551
  • C𝐴=1131525225331
  • D𝐴=1171523223551
  • E𝐴=1131523223551

Q6:

Find the multiplicative inverse of the matrix 500050005.

  • A1125250002500025
  • B250002500025
  • C1125250002500025
  • D250002500025

Q7:

Consider the matrix 𝐴=123014001. Find its inverse, given that it has the form 𝐴=1𝑝𝑞01𝑟001, where 𝑝, 𝑞, and 𝑟 are numbers that you should find.

  • A𝐴=125014001
  • B𝐴=123014001
  • C𝐴=133015001
  • D𝐴=125014011
  • E𝐴=125014001

Q8:

Consider the matrix 𝐴=2140530010. Find its inverse, given that it has the form 𝐴=𝑋𝑝𝑞0𝑌𝑟00𝑍, where 𝑋, 𝑌, 𝑍, 𝑝, 𝑞, and 𝑟 are numbers you should find.

  • A𝐴=141120161300110
  • B𝐴=141120131500110
  • C𝐴=121140151300110
  • D𝐴=121101710001535000110
  • E𝐴=121130161300110

Q9:

Consider the matrix 𝐴=1𝑎𝑏01𝑐001. Find its inverse, given that it has the form 𝐴=1𝑝𝑞01𝑟001, where 𝑝, 𝑞, and 𝑟 are expressions involving 𝑎, 𝑏, and 𝑐 that you should find.

  • A𝐴=1𝑎𝑎𝑐𝑏01𝑐001
  • B𝐴=1𝑎𝑎𝑐𝑏01𝑐001
  • C𝐴=1𝑎𝑎𝑐01𝑐001
  • D𝐴=1𝑎𝑏01𝑐001
  • E𝐴=1𝑎𝑎𝑐01𝑐001

Q10:

Consider the matrix 𝐴=𝐾𝑎𝑏0𝐿𝑐00𝑀.

Find its inverse, given that it has the form 𝐴=𝑋𝑝𝑞0𝑌𝑟00𝑍, where 𝑋, 𝑌, 𝑍, 𝑝, 𝑞, and 𝑟 are expressions involving 𝐾, 𝐿, 𝑀, 𝑎, 𝑏, and 𝑐 that you should find.

  • A𝐴=𝐿𝑀𝑎𝑀(𝑎𝑐+𝑏𝐿)0𝐾𝑀𝑐𝐾00𝐾𝐿,𝐾𝐿𝑀0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • B𝐴=1𝑎𝐿(𝑎𝑐𝑏𝐿)𝐿𝑀0𝐾𝐿𝑐𝐾𝐿𝑀00𝐾𝑀,𝐿𝑀0 (i.e., neither 𝐿 nor 𝑀 is zero)
  • C𝐴=1𝐾𝑎𝐾𝐿(𝑎𝑐𝑏𝐿)𝐾𝐿𝑀01𝐿𝑐𝐿𝑀001𝑀,𝐾𝐿𝑀0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • D𝐴=𝐿𝑀𝑎𝑀(𝑎𝑐𝑏𝐿)0𝐾𝑀𝑐𝐾00𝐾𝐿,𝐾𝐿𝑀0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • E𝐴=1𝐾𝑎𝐾𝐿(𝑎𝑐+𝑏𝐿)𝐾𝐿𝑀01𝐿𝑐𝐿𝑀001𝑀,𝐾𝐿𝑀0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)

Q11:

Using elementary row operations, find 𝐴, if possible, for the matrix 𝐴=3585123124721353.

  • A𝐴=2523321502101101
  • B𝐴=3215252302101101
  • CThe matrix has no inverse.
  • D𝐴=143215252302101101
  • E𝐴=1432152166208401211

Q12:

Find the cofactor matrix of 𝐴=758372048.

  • A48846245638122864
  • B758372048
  • C48241285628463864
  • D48241285628463864

Q13:

Given that 𝐴=587601548, determine the value of 𝐴.

Q14:

Find, if it exists, the inverse of the matrix 120021311.

  • A172727371717675727
  • B173767271757271727
  • C15356525151251525
  • D15252535151565125
  • E173767271757271727

Q15:

Determine whether the matrix 133241011 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.

  • AIt has an inverse, which is 103231353231323.
  • BIt has an inverse, which is 34121201414945412.
  • CIt has an inverse, which is 34094121454121412.
  • DIt has an inverse, which is 123230131335323.
  • EThe matrix has no inverse.

Q16:

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 123212329212010.
  • CIt has an inverse, which is 123203292112120.
  • DIt has an inverse, which is 131391020.
  • EIt has an inverse, which is 130392110.

Q17:

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.
  • B824412412
  • C824412412
  • D844211422
  • E844211422

Q18:

If 𝐴 is a square matrix and |𝐴|=18, what is 𝐴×(𝐴)adj?

  • A118𝐼
  • B18𝐼
  • C18
  • D𝐼

Q19:

Using the formula for the inverse in terms of the cofactor matrix, find the inverse of the matrix 𝑒000𝑒𝑡𝑒𝑡0𝑒𝑡𝑒𝑡𝑒𝑡+𝑒𝑡.cossincossincossin

  • A𝑒000𝑒(𝑡+𝑡)𝑒𝑡0𝑒(𝑡𝑡)𝑒𝑡cossinsincossincos
  • B𝑒000𝑒(𝑡+𝑡)𝑒𝑡0𝑒(𝑡𝑡)𝑒𝑡cossinsincossincos
  • C𝑒000𝑒(𝑡+𝑡)𝑒𝑡𝑡0𝑒𝑡𝑒𝑡cossincossinsincos
  • D𝑒000𝑒(𝑡+𝑡)𝑒𝑡0𝑒(𝑡𝑡)𝑒𝑡cossinsincossincos
  • E𝑒000𝑒(𝑡+𝑡)𝑒𝑡0𝑒(𝑡𝑡)𝑒𝑡cossinsincossincos

Q20:

Find the adjoint matrix of the matrix 𝐴=272979854.

  • A73187736801014677
  • B73187736801014677
  • C73361011884677077
  • D272979854

Q21:

Is there any value of 𝑡 for which the matrix 𝑒𝑒𝑡𝑒𝑡𝑒𝑒𝑡𝑒𝑡𝑒𝑡+𝑒𝑡𝑒2𝑒𝑡2𝑒𝑡cossincossinsincossincos has no inverse?

  • Ayes, when 𝑡=1
  • Byes, when 𝑡=0
  • Cyes, when 𝑡=2
  • Dyes, when 𝑡=1
  • Eno

Q22:

Find the value of 𝑥 that makes the matrix 133𝑥3𝑥𝑥+1555 singular.

Q23:

For the matrix 1𝑡𝑡012𝑡𝑡02, does there exist a value of 𝑡 for which it fails to have an inverse? if so, what is this value?

  • Ayes, when 𝑡=2.
  • Byes, when 𝑡=23.
  • Cyes, when 𝑡=23.
  • Dyes, when 𝑡=2.
  • Eyes, when 𝑡=1.

Q24:

Does the matrix 𝐴=545090272 have a multiplicative inverse?

  • AYes
  • BNo

Q25:

Find the set of real values of 𝑥 that make the matrix 𝑥4313311𝑥54 singular.

  • A{7,7}
  • B{8,6}
  • C{6,8}
  • D{5,5}

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