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Worksheet: Finding the Inverse of a Matrix Using the Adjoint Matrix Method

In this worksheet, we will practice determining whether a matrix has an inverse and then finding it using the adjoint matrix method.

Q1:

Find, if it exists, the inverse of the matrix

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

Q2:

Determine whether the matrix 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 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 3 4 0 − 9 4 − 1 2 1 4 5 4 1 2 − 1 4 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
  • BIt has an inverse, which is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 − 2 3 2 3 0 1 3 − 1 3 − 3 5 3 − 2 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
  • CIt has an inverse, which is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 3 4 − 1 2 1 2 0 1 4 − 1 4 − 9 4 5 4 − 1 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
  • DIt has an inverse, which is ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 − 3 − 2 3 1 3 5 3 2 3 − 1 3 − 2 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .
  • E The matrix has no inverse.

Q3:

Consider the matrix 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 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ − 1 2 3 2 1 2 3 2 − 9 2 − 1 2 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .
  • BThere is no inverse because its determinant equals zero.
  • CIt has an inverse, which is − 1 3 1 3 − 9 − 1 0 2 0 .
  • DIt has an inverse, which is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − 1 2 3 2 0 3 2 − 9 2 1 1 2 − 1 2 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
  • EIt has an inverse, which is − 1 3 0 3 − 9 2 1 − 1 0 .

Q4:

Determine whether the matrix 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
  • B 8 − 2 − 4 − 4 1 2 − 4 1 2
  • C 8 4 − 4 2 1 − 1 − 4 − 2 2
  • DThere is no inverse because the determinant equals zero.
  • E 8 2 − 4 4 1 − 2 − 4 − 1 2

Q5:

Which of the below matrices does NOT have an inverse?

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

Q6:

If 𝐴 is a square matrix and | 𝐴 | = − 1 8 , what is 𝐴 × ( 𝐴 ) a d j ?

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

Q7:

Using the formula for the inverse in terms of the cofactor matrix, find the inverse of the matrix

  • A 𝑒 0 0 0 𝑒 ( 𝑡 + 𝑡 ) 𝑒 𝑡 0 𝑒 ( 𝑡 − 𝑡 ) 𝑒 𝑡 − 𝑡 − 𝑡 − 𝑡 − 𝑡 − 𝑡 c o s s i n s i n c o s s i n c o s
  • B 𝑒 0 0 0 𝑒 ( 𝑡 + 𝑡 ) 𝑒 𝑡 0 𝑒 ( 𝑡 − 𝑡 ) 𝑒 𝑡 2 𝑡 2 𝑡 2 𝑡 2 𝑡 2 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 2 𝑡 2 𝑡 2 𝑡 2 𝑡 2 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 − 𝑡 − 𝑡 − 𝑡 − 𝑡 − 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 − 𝑡 0 − 𝑒 𝑡 𝑒 𝑡 − 2 𝑡 − 𝑡 − 𝑡 − 𝑡 − 𝑡 c o s s i n c o s s i n s i n c o s

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