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

Q2:

Find the inverse of the matrix 𝑒𝑡𝑡𝑒𝑡𝑡𝑒𝑡𝑡.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=332161312.

  • 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=224111256.

  • 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=110103052.

  • 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 500050005.

  • 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=758372048.

  • 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=587601548, 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 123230131335323.
  • B The matrix has no inverse.
  • CIt has an inverse, which is 34121201414945412.
  • DIt has an inverse, which is 34094121454121412.
  • EIt has an inverse, which is 103231353231323.

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 130392110.
  • BIt has an inverse, which is 123203292112120.
  • CThere is no inverse because its determinant equals zero.
  • DIt has an inverse, which is 123212329212010.
  • EIt has an inverse, which is 131391020.

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 𝑒000𝑒𝑡𝑒𝑡0𝑒𝑡𝑒𝑡𝑒𝑡+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=272979854.

  • 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 𝑒𝑒𝑡𝑒𝑡𝑒𝑒𝑡𝑒𝑡𝑒𝑡+𝑒𝑡𝑒2𝑒𝑡2𝑒𝑡cossincossinsincossincos 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 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?

  • 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=545090272 have a multiplicative inverse?

  • ANo
  • BYes

Q25:

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

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

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