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Worksheet: Finding the Inverse of a 3x3 Matrix

Q1:

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

  • AIt has an inverse, 1 2 5 3 2 5 4 2 5 3 2 5 9 2 5 1 2 5 6 2 5 1 5 2 2 5 .
  • BIt has an inverse, 1 1 3 3 1 3 6 1 3 3 1 3 9 1 3 5 1 3 4 1 3 1 1 3 2 1 3 .
  • CIt has an inverse, 1 2 5 3 2 5 6 2 5 3 2 5 9 2 5 1 5 4 2 5 1 2 5 2 2 5 .
  • DIt has an inverse, 1 1 3 3 1 3 4 1 3 3 1 3 9 1 3 1 1 3 6 1 3 5 1 3 2 1 3 .
  • EIt has no inverse.

Q2:

Find the inverse of the matrix

  • 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 ) 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 0 𝑒 ( 𝑡 + 𝑡 ) 2 𝑡 ( 𝑡 𝑡 ) ( 𝑡 𝑡 ) 2 𝑡 ( 𝑡 + 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
  • C 𝑒 ( 𝑡 + 𝑡 ) ( 𝑡 𝑡 ) 0 2 𝑡 2 𝑡 𝑒 ( 𝑡 𝑡 ) ( 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
  • 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 ) 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 1 2 𝑒 0 1 2 𝑒 1 2 ( 𝑡 + 𝑡 ) 𝑡 𝑡 1 2 ( 𝑡 𝑡 ) 𝑡 1 2 ( 𝑡 + 𝑡 ) 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

Q3:

Use technology to find the inverse of the matrix

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

Q4:

Use technology to find the inverse of the matrix

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

Q5:

Use technology to find the inverse of the matrix

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

Q6:

Using the elementary row operation, find 𝐴 1 , if possible, for the matrix

  • A 𝐴 = 1 4 6 0 2 1 3 2 5 1 7 2 5 1
  • B 𝐴 = 1 4 6 0 2 1 3 2 5 1 7 2 5 1
  • C 𝐴 = 1 4 6 0 2 1 3 2 5 1 7 2 5 1
  • D 𝐴 = 1 4 6 0 2 1 3 2 5 1 7 2 5 1
  • E The matrix has no inverse.

Q7:

Using the Cayley-Hamilton theorem, find 𝐴 1 , if possible, for the matrix

  • A 𝐴 = 1 5 6 7 3 5 2 8 7 1 0 3 2 7 5 5 2 5 1
  • B 𝐴 = 1 4 5 3 8 7 7 5 5 1 0 5 2 3 2 2 5 1
  • C 𝐴 = 1 4 5 3 5 2 8 7 1 0 3 2 7 5 5 2 5 1
  • D 𝐴 = 1 4 5 3 5 2 8 7 1 0 3 2 7 5 5 2 5 1
  • EThe matrix has no inverse.

Q8:

Using elementary row operations, find 𝐴 1 for the matrix if possible.

  • A 𝐴 = 1 3 9 6 3 6 4 3 9 5 3 1
  • B 𝐴 = 1 3 9 6 3 6 4 3 9 5 3 1
  • C 𝐴 = 9 6 3 6 4 3 9 5 3 1
  • D The matrix has no inverse.
  • E 𝐴 = 1 3 9 6 3 6 4 3 9 5 3 1

Q9:

Find the multiplicative inverse of the following.

  • A
  • B
  • C
  • D

Q10:

Find the multiplicative inverse of the following matrix.

  • A
  • B
  • C
  • D

Q11:

Find the multiplicative inverse of the following matrix.

  • A
  • B
  • C
  • D

Q12:

Consider the matrix 𝐴 = 1 2 3 0 1 4 0 0 1 . Find its inverse, given that it has the form 𝐴 = 1 𝑝 𝑞 0 1 𝑟 0 0 1 1 , where 𝑝 , 𝑞 , and 𝑟 are numbers that you should find.

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

Q13:

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

  • A 𝐴 = 1 2 1 1 3 0 1 6 1 3 0 0 1 1 0 1
  • B 𝐴 = 1 2 1 1 4 0 1 5 1 3 0 0 1 1 0 1
  • C 𝐴 = 1 4 1 1 2 0 1 3 1 5 0 0 1 1 0 1
  • D 𝐴 = 1 2 1 1 0 1 7 1 0 0 0 1 5 3 5 0 0 0 1 1 0 1
  • E 𝐴 = 1 4 1 1 2 0 1 6 1 3 0 0 1 1 0 1

Q14:

Consider the shown matrices. Find the matrix .

  • A
  • B
  • C
  • D

Q15:

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

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

Q16:

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

  • A 𝐴 = 𝐿 𝑀 𝑎 𝑀 ( 𝑎 𝑐 𝑏 𝐿 ) 0 𝐾 𝑀 𝑐 𝐾 0 0 𝐾 𝐿 , 𝐾 𝐿 𝑀 0 1 (i.e., none of 𝐾 , 𝐿 , and 𝑀 is zero.)
  • B 𝐴 = 1 𝑎 𝐿 ( 𝑎 𝑐 𝑏 𝐿 ) 𝐿 𝑀 0 𝐾 𝐿 𝑐 𝐾 𝐿 𝑀 0 0 𝐾 𝑀 , 𝐿 𝑀 0 1 (i.e., neither 𝐿 nor 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 1 (i.e., none of 𝐾 , 𝐿 , and 𝑀 is zero.)
  • D 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 1 (i.e., none of 𝐾 , 𝐿 , and 𝑀 is zero.)
  • E 𝐴 = 𝐿 𝑀 𝑎 𝑀 ( 𝑎 𝑐 + 𝑏 𝐿 ) 0 𝐾 𝑀 𝑐 𝐾 0 0 𝐾 𝐿 , 𝐾 𝐿 𝑀 0 1 (i.e., none of 𝐾 , 𝐿 , and 𝑀 is zero.)