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 an inverse, ⎡⎢⎢⎢⎢⎢⎣125−325625−325925−15425125−225⎤⎥⎥⎥⎥⎥⎦.
  • BIt has an inverse, ⎡⎢⎢⎢⎢⎢⎣125−325425−325925125625−15−225⎤⎥⎥⎥⎥⎥⎦.
  • CIt has an inverse, ⎡⎢⎢⎢⎢⎢⎣113−313413−313913113613−513−213⎤⎥⎥⎥⎥⎥⎦.
  • DIt has no inverse.
  • EIt has an inverse, ⎡⎢⎢⎢⎢⎢⎣113−313613−313913−513413113−213⎤⎥⎥⎥⎥⎥⎦.

Q2:

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

  • A ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 𝑒 1 2 ( 𝑡 + 𝑡 ) 1 2 ( 𝑡 − 𝑡 ) 0 − 𝑡 𝑡 1 2 𝑒 1 2 ( 𝑡 − 𝑡 ) − 1 2 ( 𝑡 + 𝑡 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ s i n c o s s i n c o s s i n c o s s i n c o s s i n c o s
  • B 𝑒 ( 𝑡 + 𝑡 ) ( 𝑡 − 𝑡 ) 0 − 2 𝑡 2 𝑡 𝑒 ( 𝑡 − 𝑡 ) − ( 𝑡 + 𝑡 ) s i n c o s s i n c o s s i n c o s s i n c o s s i n c o s
  • C ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 𝑒 0 1 2 𝑒 1 2 ( 𝑡 + 𝑡 ) − 𝑡 𝑡 1 2 ( 𝑡 − 𝑡 ) 𝑡 − 1 2 ( 𝑡 + 𝑡 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ s i n c o s s i n c o s s i n c o s c o s s i n c o s
  • D ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 𝑒 0 1 2 𝑒 1 2 ( 𝑡 + 𝑡 ) − 𝑡 1 2 ( 𝑡 − 𝑡 ) 1 2 ( 𝑡 − 𝑡 ) 𝑡 − 1 2 ( 𝑡 + 𝑡 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ s i n c o s s i n s i n c o s s i n c o s c o s s i n c o s
  • E 𝑒 0 𝑒 ( 𝑡 + 𝑡 ) − 2 𝑡 ( 𝑡 − 𝑡 ) ( 𝑡 − 𝑡 ) 2 𝑡 − ( 𝑡 + 𝑡 ) s i n c o s s i n s i n c o s s i n c o s c o s s i n c o s

Q3:

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

  • A 𝐴 = 1 7 4 − 1 3 4 − 1 5 − 1 − 1 2 5 − 1 9 1 2 1 5
  • B 𝐴 = 1 7 4 1 3 − 4 1 5 1 1 2 − 5 1 9 − 1 2 − 1 5
  • C 𝐴 = 1 8 0 − 1 3 4 − 1 5 − 1 − 1 2 5 − 1 9 1 2 1 5
  • D 𝐴 = 1 8 0 1 3 1 1 9 − 4 1 2 − 1 2 1 5 − 5 − 1 5
  • E 𝐴 = 1 8 0 1 3 − 4 1 5 1 1 2 − 5 1 9 − 1 2 − 1 5

Q4:

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

  • A 𝐴 = 1 2 2 − 1 8 2 4 4 − 2 − 3 − 6 0
  • B 𝐴 = 1 6 1 − 4 3 − 8 − 4 6 − 2 2 0
  • C 𝐴 = 1 6 − 1 8 2 4 4 − 2 − 3 − 6 0
  • D 𝐴 = 1 6 1 − 8 − 2 − 4 − 4 2 3 6 0
  • E 𝐴 = 1 2 2 1 − 8 − 2 − 4 − 4 2 3 6 0

Q5:

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

  • A 𝐴 = 1 1 7 1 5 − 2 − 3 − 2 2 3 − 5 5 1
  • B 𝐴 = 1 1 3 1 5 − 2 − 3 − 2 2 3 − 5 5 1
  • C 𝐴 = 1 1 3 1 5 − 2 − 5 − 2 2 5 − 3 3 1
  • D 𝐴 = 1 1 3 − 1 5 2 3 2 − 2 − 3 5 − 5 − 1
  • E 𝐴 = 1 1 7 − 1 5 2 3 2 − 2 − 3 5 − 5 − 1

Q6:

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

  • A − 1 1 2 5 − 2 5 0 0 0 − 2 5 0 0 0 − 2 5
  • B − 1 1 2 5 2 5 0 0 0 2 5 0 0 0 2 5
  • C 2 5 0 0 0 2 5 0 0 0 2 5
  • D − 2 5 0 0 0 − 2 5 0 0 0 − 2 5

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.

  • A 𝐴 = 1 − 2 − 3 0 1 − 4 0 0 1
  • B 𝐴 = 1 2 − 5 0 1 4 0 0 1
  • C 𝐴 = 1 3 − 3 0 1 5 0 0 1
  • D 𝐴 = 1 2 − 5 0 1 − 4 0 1 1
  • E 𝐴 = 1 − 2 5 0 1 − 4 0 0 1

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.

  • A 𝐴 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 − 1 1 0 − 1 7 1 0 0 0 1 5 − 3 5 0 0 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • B 𝐴 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 − 1 1 3 0 1 6 1 3 0 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • C 𝐴 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 4 − 1 1 2 0 1 6 1 3 0 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • D 𝐴 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 1 1 4 0 1 5 1 3 0 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • E 𝐴 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 4 1 1 2 0 1 3 1 5 0 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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.

  • A 𝐴 = 1 𝑎 𝑎 𝑐 0 1 𝑐 0 0 1
  • B 𝐴 = 1 − 𝑎 𝑎 𝑐 − 𝑏 0 1 − 𝑐 0 0 1
  • C 𝐴 = 1 − 𝑎 − 𝑏 0 1 − 𝑐 0 0 1
  • D 𝐴 = 1 𝑎 𝑎 𝑐 − 𝑏 0 1 𝑐 0 0 1
  • E 𝐴 = 1 − 𝑎 𝑎 𝑐 0 1 − 𝑐 0 0 1

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.

  • A 𝐴 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 − 𝑎 𝐿 ( 𝑎 𝑐 − 𝑏 𝐿 ) 𝐿 𝑀 0 𝐾 𝐿 − 𝑐 𝐾 𝐿 𝑀 0 0 𝐾 𝑀 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐿 𝑀 ≠ 0 (i.e., neither 𝐿 nor 𝑀 is zero)
  • B 𝐴 = 𝐿 𝑀 − 𝑎 𝑀 ( 𝑎 𝑐 − 𝑏 𝐿 ) 0 𝐾 𝑀 − 𝑐 𝐾 0 0 𝐾 𝐿 , 𝐾 𝐿 𝑀 ≠ 0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • C 𝐴 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 𝐾 − 𝑎 𝐾 𝐿 ( 𝑎 𝑐 − 𝑏 𝐿 ) 𝐾 𝐿 𝑀 0 1 𝐿 − 𝑐 𝐿 𝑀 0 0 1 𝑀 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐾 𝐿 𝑀 ≠ 0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • D 𝐴 = 𝐿 𝑀 − 𝑎 𝑀 ( 𝑎 𝑐 + 𝑏 𝐿 ) 0 𝐾 𝑀 − 𝑐 𝐾 0 0 𝐾 𝐿 , 𝐾 𝐿 𝑀 ≠ 0 (i.e., none of 𝐾,𝐿, and 𝑀 is zero)
  • E 𝐴 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 𝐾 − 𝑎 𝐾 𝐿 ( 𝑎 𝑐 + 𝑏 𝐿 ) 𝐾 𝐿 𝑀 0 1 𝐿 − 𝑐 𝐿 𝑀 0 0 1 𝑀 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐾 𝐿 𝑀 ≠ 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⎤⎥⎥⎦.

  • A 𝐴 = ⎡ ⎢ ⎢ ⎣ 3 − 2 1 − 5 − 2 5 − 2 3 0 − 2 1 0 − 1 1 0 1 ⎤ ⎥ ⎥ ⎦
  • B The matrix has no inverse.
  • C 𝐴 = 1 4 ⎡ ⎢ ⎢ ⎣ 3 − 2 1 − 5 − 2 5 − 2 3 0 2 − 1 0 1 − 1 0 − 1 ⎤ ⎥ ⎥ ⎦
  • D 𝐴 = ⎡ ⎢ ⎢ ⎣ − 2 5 − 2 3 3 − 2 1 − 5 0 − 2 1 0 1 − 1 0 − 1 ⎤ ⎥ ⎥ ⎦
  • E 𝐴 = 1 4 ⎡ ⎢ ⎢ ⎣ 3 − 2 1 − 5 − 2 1 6 − 6 2 0 − 8 4 0 1 − 2 − 1 1 ⎤ ⎥ ⎥ ⎦

Q12:

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

  • A 7 5 − 8 3 − 7 2 0 4 − 8
  • B 4 8 2 4 1 2 8 − 5 6 − 2 8 − 4 6 − 3 8 − 6 4
  • C 4 8 − 2 4 1 2 − 8 − 5 6 2 8 − 4 6 3 8 − 6 4
  • D 4 8 − 8 − 4 6 − 2 4 − 5 6 3 8 1 2 2 8 − 6 4

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 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − 1 5 2 5 − 2 5 − 3 5 − 1 5 1 5 6 5 − 1 − 2 5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 7 3 7 − 6 7 − 2 7 1 7 5 7 2 7 − 1 7 2 7 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 7 − 2 7 2 7 3 7 1 7 − 1 7 − 6 7 5 7 2 7 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − 1 5 − 3 5 6 5 2 5 − 1 5 − 1 − 2 5 1 5 − 2 5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 7 3 7 − 6 7 − 2 7 1 7 5 7 2 7 − 1 7 2 7 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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 ⎡⎢⎢⎢⎢⎢⎣1−2323013−13−353−23⎤⎥⎥⎥⎥⎥⎦.
  • B The matrix has no inverse.
  • CIt has an inverse, which is ⎡⎢⎢⎢⎢⎢⎣34−1212014−14−9454−12⎤⎥⎥⎥⎥⎥⎦.
  • DIt has an inverse, which is ⎡⎢⎢⎢⎢⎢⎣340−94−12145412−14−12⎤⎥⎥⎥⎥⎥⎦.
  • EIt has an inverse, which is ⎡⎢⎢⎢⎢⎣10−3−23135323−13−23⎤⎥⎥⎥⎥⎦.

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.

  • AIt has an inverse, which is −1303−921−10.
  • BIt has an inverse, which is ⎡⎢⎢⎢⎢⎢⎣−1232032−92112−120⎤⎥⎥⎥⎥⎥⎦.
  • CThere is no inverse because its determinant equals zero.
  • DIt has an inverse, which is ⎡⎢⎢⎢⎢⎣−12321232−92−12010⎤⎥⎥⎥⎥⎦.
  • EIt has an inverse, which is −1313−9−1020.

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.

  • A 8 4 − 4 2 1 − 1 − 4 − 2 2
  • BThere is no inverse because the determinant equals zero.
  • C 8 − 4 − 4 − 2 1 1 − 4 2 2
  • D 8 2 − 4 4 1 − 2 − 4 − 1 2
  • E 8 − 2 − 4 − 4 1 2 − 4 1 2

Q18:

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

  • A − 1 8
  • B 𝐼
  • C − 1 1 8 𝐼
  • D − 1 8 𝐼

Q19:

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

  • A 𝑒 0 0 0 𝑒 ( 𝑡 + 𝑡 ) 𝑒 𝑡 − 𝑡 0 − 𝑒 𝑡 𝑒 𝑡 c o s s i n c o s s i n s i n c o s
  • B 𝑒 0 0 0 𝑒 ( 𝑡 + 𝑡 ) − 𝑒 𝑡 0 − 𝑒 ( 𝑡 − 𝑡 ) 𝑒 𝑡 c o s s i n s i n c o s s i n c o s
  • C 𝑒 0 0 0 𝑒 ( 𝑡 + 𝑡 ) 𝑒 𝑡 0 𝑒 ( 𝑡 − 𝑡 ) 𝑒 𝑡 c o s s i n s i n c o s s i n c o s
  • D 𝑒 0 0 0 𝑒 ( 𝑡 + 𝑡 ) 𝑒 𝑡 0 𝑒 ( 𝑡 − 𝑡 ) 𝑒 𝑡 c o s s i n s i n c o s s i n c o s
  • E 𝑒 0 0 0 𝑒 ( 𝑡 + 𝑡 ) − 𝑒 𝑡 0 − 𝑒 ( 𝑡 − 𝑡 ) 𝑒 𝑡 c o s s i n s i n c o s s i n c o s

Q20:

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

  • A 2 7 2 9 − 7 9 − 8 − 5 − 4
  • B 7 3 1 8 7 7 − 3 6 8 0 − 1 0 1 − 4 6 − 7 7
  • C 7 3 3 6 − 1 0 1 − 1 8 8 4 6 7 7 0 − 7 7
  • D 7 3 − 1 8 7 7 3 6 8 0 − 1 0 1 4 6 − 7 7

Q21:

Is there any value of 𝑡 for which the matrix 𝑒𝑒𝑡𝑒𝑡𝑒−𝑒𝑡−𝑒𝑡−𝑒𝑡+𝑒𝑡𝑒2𝑒𝑡−2𝑒𝑡cossincossinsincossincos has no inverse?

  • A yes, when 𝑡=−1
  • B yes, when 𝑡=0
  • C no
  • D yes, when 𝑡=2
  • E yes, when 𝑡=1

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?

  • A yes, when 𝑡=−2.
  • B yes, when 𝑡=−√2.
  • C yes, when 𝑡=−1.
  • D yes, when 𝑡=−23.
  • E yes, when 𝑡=−23.

Q24:

Does the matrix 𝐴=5−4−50−9027−2 have a multiplicative inverse?

  • ANo
  • BYes

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 { − 5 , 5 }
  • D { − 6 , 8 }

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