# 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 has an inverse. If so, find the inverse by considering the matrix of cofactors.

• AIt has no inverse.
• BIt has an inverse, .
• CIt has an inverse, .
• DIt has an inverse, .
• EIt has an inverse, .

Q2:

Find the inverse of the matrix

• A
• B
• C
• D
• E

Q3:

Use technology to find the inverse of the matrix

• A
• B
• C
• D
• E

Q4:

Use technology to find the inverse of the matrix

• A
• B
• C
• D
• E

Q5:

Use technology to find the inverse of the matrix

• A
• B
• C
• D
• E

Q6:

Find the multiplicative inverse of the matrix

• A
• B
• C
• D

Q7:

Consider the matrix Find its inverse, given that it has the form , where , , and are numbers that you should find.

• A
• B
• C
• D
• E

Q8:

Consider the matrix Find its inverse, given that it has the form where , , , , , and are numbers you should find.

• A
• B
• C
• D
• E

Q9:

Consider the matrix Find its inverse, given that it has the form where , , and are expressions involving , , and that you should find.

• A
• B
• C
• D
• E

Q10:

Consider the matrix

Find its inverse, given that it has the form where , , , , , and are expressions involving , , , , , and that you should find.

• A (i.e., none of and is zero)
• B (i.e., neither nor is zero)
• C (i.e., none of and is zero)
• D (i.e., none of and is zero)
• E (i.e., none of and is zero)

Q11:

Using elementary row operations, find , if possible, for the matrix

• A
• B
• CThe matrix has no inverse.
• D
• E

Q12:

Find the cofactor matrix of

• A
• B
• C
• D

Q13:

Given that determine the value of .

Q14:

Find, if it exists, the inverse of the matrix

• A
• B
• C
• D
• E

Q15:

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 .
• BIt has an inverse, which is .
• CIt has an inverse, which is .
• DIt has an inverse, which is .
• EThe matrix has no inverse.

Q16:

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.

• AThere is no inverse because its determinant equals zero.
• BIt has an inverse, which is .
• CIt has an inverse, which is .
• DIt has an inverse, which is .
• EIt has an inverse, which is .

Q17:

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.

• AThere is no inverse because the determinant equals zero.
• B
• C
• D
• E

Q18:

If is a square matrix and , what is ?

• A
• B
• C
• D

Q19:

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

• A
• B
• C
• D
• E

Q20:

Find the adjoint matrix of the matrix

• A
• B
• C
• D

Q21:

Is there any value of for which the matrix has no inverse?

• Ayes, when
• Byes, when
• Cyes, when
• Dyes, when
• Eno

Q22:

Find the value of that makes the matrix singular.

Q23:

For the matrix does there exist a value of for which it fails to have an inverse? if so, what is this value?

• Ayes, when .
• Byes, when .
• Cyes, when .
• Dyes, when .
• Eyes, when .

Q24:

Does the matrix have a multiplicative inverse?

• AYes
• BNo

Q25:

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

• A
• B
• C
• D