Worksheet: Inverse of a Matrix: The Adjoint Method

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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, ⎡⎢⎢⎢⎢⎢⎣113−313413−313913113613−513−213⎤⎥⎥⎥⎥⎥⎦.
  • CIt has an inverse, ⎡⎢⎢⎢⎢⎢⎣113−313613−313913−513413113−213⎤⎥⎥⎥⎥⎥⎦.
  • DIt has an inverse, ⎡⎢⎢⎢⎢⎢⎣125−325625−325925−15425125−225⎤⎥⎥⎥⎥⎥⎦.
  • EIt has an inverse, ⎡⎢⎢⎢⎢⎢⎣125−325425−325925125625−15−225⎤⎥⎥⎥⎥⎥⎦.

Q2:

Find the inverse of the matrix 𝑒𝑡𝑡𝑒−𝑡𝑡𝑒−𝑡−𝑡.cossinsincoscossin

  • A𝑒0𝑒(𝑡+𝑡)−2𝑡(𝑡−𝑡)(𝑡−𝑡)2𝑡−(𝑡+𝑡)sincossinsincossincoscossincos
  • B⎡⎢⎢⎢⎢⎢⎣12𝑒012𝑒12(𝑡+𝑡)−𝑡𝑡12(𝑡−𝑡)𝑡−12(𝑡+𝑡)⎤⎥⎥⎥⎥⎥⎦sincossincossincoscossincos
  • C⎡⎢⎢⎢⎢⎢⎣12𝑒012𝑒12(𝑡+𝑡)−𝑡12(𝑡−𝑡)12(𝑡−𝑡)𝑡−12(𝑡+𝑡)⎤⎥⎥⎥⎥⎥⎦sincossinsincossincoscossincos
  • D⎡⎢⎢⎢⎢⎣12𝑒12(𝑡+𝑡)12(𝑡−𝑡)0−𝑡𝑡12𝑒12(𝑡−𝑡)−12(𝑡+𝑡)⎤⎥⎥⎥⎥⎦sincossincossincossincossincos
  • E𝑒(𝑡+𝑡)(𝑡−𝑡)0−2𝑡2𝑡𝑒(𝑡−𝑡)−(𝑡+𝑡)sincossincossincossincossincos

Q3:

Use technology to find the inverse of the matrix 𝐴=33216−13−1−2.

  • A𝐴=17413−415112−519−12−15
  • B𝐴=180−134−15−1−125−191215
  • C𝐴=18013119−412−1215−5−15
  • D𝐴=18013−415112−519−12−15
  • E𝐴=174−134−15−1−125−191215

Q4:

Use technology to find the inverse of the matrix 𝐴=224−1−1−1256.

  • A𝐴=161−43−8−46−220
  • B𝐴=16−18244−2−3−60
  • C𝐴=1221−8−2−4−42360
  • D𝐴=161−8−2−4−42360
  • E𝐴=122−18244−2−3−60

Q5:

Use technology to find the inverse of the matrix 𝐴=11010305−2.

  • A𝐴=117−15232−2−35−5−1
  • B𝐴=113−15232−2−35−5−1
  • C𝐴=11315−2−5−225−331
  • D𝐴=11715−2−3−223−551
  • E𝐴=11315−2−3−223−551

Q6:

Find the multiplicative inverse of the matrix −5000−5000−5.

  • A−1125250002500025
  • B250002500025
  • C−1125−25000−25000−25
  • D−25000−25000−25

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=12−5014001
  • B𝐴=1−2−301−4001
  • C𝐴=13−3015001
  • D𝐴=12−501−4011
  • E𝐴=1−2501−4001

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=⎡⎢⎢⎢⎢⎢⎣14−1120161300110⎤⎥⎥⎥⎥⎥⎦
  • B𝐴=⎡⎢⎢⎢⎢⎢⎣141120131500110⎤⎥⎥⎥⎥⎥⎦
  • C𝐴=⎡⎢⎢⎢⎢⎢⎣121140151300110⎤⎥⎥⎥⎥⎥⎦
  • D𝐴=⎡⎢⎢⎢⎢⎢⎣12−110−17100015−35000110⎤⎥⎥⎥⎥⎥⎦
  • E𝐴=⎡⎢⎢⎢⎢⎢⎣12−1130161300110⎤⎥⎥⎥⎥⎥⎦

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=1−𝑎𝑎𝑐−𝑏01−𝑐001
  • B𝐴=1𝑎𝑎𝑐−𝑏01𝑐001
  • C𝐴=1−𝑎𝑎𝑐01−𝑐001
  • D𝐴=1−𝑎−𝑏01−𝑐001
  • 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=𝐿𝑀−𝑎𝑀(𝑎𝑐+𝑏𝐿)0𝐾𝑀−𝑐𝐾00𝐾𝐿,𝐾𝐿𝑀≠0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • B𝐴=⎡⎢⎢⎢⎢⎢⎣1−𝑎𝐿(𝑎𝑐−𝑏𝐿)𝐿𝑀0𝐾𝐿−𝑐𝐾𝐿𝑀00𝐾𝑀⎤⎥⎥⎥⎥⎥⎦,𝐿𝑀≠0 (i.e., neither 𝐿 nor 𝑀 is zero)
  • C𝐴=⎡⎢⎢⎢⎢⎢⎣1𝐾−𝑎𝐾𝐿(𝑎𝑐−𝑏𝐿)𝐾𝐿𝑀01𝐿−𝑐𝐿𝑀001𝑀⎤⎥⎥⎥⎥⎥⎦,𝐾𝐿𝑀≠0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • D𝐴=𝐿𝑀−𝑎𝑀(𝑎𝑐−𝑏𝐿)0𝐾𝑀−𝑐𝐾00𝐾𝐿,𝐾𝐿𝑀≠0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • E𝐴=⎡⎢⎢⎢⎢⎢⎣1𝐾−𝑎𝐾𝐿(𝑎𝑐+𝑏𝐿)𝐾𝐿𝑀01𝐿−𝑐𝐿𝑀001𝑀⎤⎥⎥⎥⎥⎥⎦,𝐾𝐿𝑀≠0 (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=⎡⎢⎢⎣−25−233−21−50−2101−10−1⎤⎥⎥⎦
  • B𝐴=⎡⎢⎢⎣3−21−5−25−230−210−1101⎤⎥⎥⎦
  • CThe matrix has no inverse.
  • D𝐴=14⎡⎢⎢⎣3−21−5−25−2302−101−10−1⎤⎥⎥⎦
  • E𝐴=14⎡⎢⎢⎣3−21−5−216−620−8401−2−11⎤⎥⎥⎦

Q12:

Find the cofactor matrix of 𝐴=7−5−8−3−7−20−4−8.

  • A48−8−46−24−56381228−64
  • B75−83−7204−8
  • C48−2412−8−5628−4638−64
  • D4824128−56−28−46−38−64

Q13:

Given that 𝐴=−58−76015−4−8, determine the value of 𝐴.

Q14:

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

  • A⎡⎢⎢⎢⎢⎢⎣17−27273717−17−675727⎤⎥⎥⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎢⎢⎣1737−67−27175727−1727⎤⎥⎥⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎢⎢⎣−15−356525−15−1−2515−25⎤⎥⎥⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎢⎢⎣−1525−25−35−151565−1−25⎤⎥⎥⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎢⎢⎣1737−67−27175727−1727⎤⎥⎥⎥⎥⎥⎦

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 ⎡⎢⎢⎢⎢⎣10−3−23135323−13−23⎤⎥⎥⎥⎥⎦.
  • BIt has an inverse, which is ⎡⎢⎢⎢⎢⎢⎣34−1212014−14−9454−12⎤⎥⎥⎥⎥⎥⎦.
  • CIt has an inverse, which is ⎡⎢⎢⎢⎢⎢⎣340−94−12145412−14−12⎤⎥⎥⎥⎥⎥⎦.
  • DIt has an inverse, which is ⎡⎢⎢⎢⎢⎢⎣1−2323013−13−353−23⎤⎥⎥⎥⎥⎥⎦.
  • 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 ⎡⎢⎢⎢⎢⎣−12321232−92−12010⎤⎥⎥⎥⎥⎦.
  • CIt has an inverse, which is ⎡⎢⎢⎢⎢⎢⎣−1232032−92112−120⎤⎥⎥⎥⎥⎥⎦.
  • DIt has an inverse, which is −1313−9−1020.
  • EIt has an inverse, which is −1303−921−10.

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.
  • B8−2−4−412−412
  • C82−441−2−4−12
  • D8−4−4−211−422
  • E84−421−1−4−22

Q18:

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

  • A−118𝐼
  • B−18𝐼
  • C−18
  • D𝐼

Q19:

Using the formula for the inverse in terms of the cofactor matrix, find the inverse of the matrix 𝑒000𝑒𝑡𝑒𝑡0𝑒𝑡−𝑒𝑡𝑒𝑡+𝑒𝑡.cossincossincossin

  • A𝑒000𝑒(𝑡+𝑡)𝑒𝑡0𝑒(𝑡−𝑡)𝑒𝑡cossinsincossincos
  • B𝑒000𝑒(𝑡+𝑡)𝑒𝑡0𝑒(𝑡−𝑡)𝑒𝑡cossinsincossincos
  • C𝑒000𝑒(𝑡+𝑡)𝑒𝑡−𝑡0−𝑒𝑡𝑒𝑡cossincossinsincos
  • D𝑒000𝑒(𝑡+𝑡)−𝑒𝑡0−𝑒(𝑡−𝑡)𝑒𝑡cossinsincossincos
  • E𝑒000𝑒(𝑡+𝑡)−𝑒𝑡0−𝑒(𝑡−𝑡)𝑒𝑡cossinsincossincos

Q20:

Find the adjoint matrix of the matrix 𝐴=2−72−9−7−9−85−4.

  • A73−18773680−10146−77
  • B731877−3680−101−46−77
  • C7336−101−18846770−77
  • D2729−79−8−5−4

Q21:

Is there any value of 𝑡 for which the matrix 𝑒𝑒𝑡𝑒𝑡𝑒−𝑒𝑡−𝑒𝑡−𝑒𝑡+𝑒𝑡𝑒2𝑒𝑡−2𝑒𝑡cossincossinsincossincos 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 −13−3−𝑥−3𝑥𝑥+1−5−5−5 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=5−4−50−9027−2 have a multiplicative inverse?

  • AYes
  • BNo

Q25:

Find the set of real values of 𝑥 that make the matrix 𝑥−431331−1𝑥−5−4 singular.

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

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