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Matrix inverse worksheets

Decide whether six numeric 2 × 2 matrices have inverses, then find six exact inverses or state that no inverse exists.

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For each 2 × 2 matrix, calculate ad − bc. A matrix has an inverse exactly when that determinant is nonzero. For the final six questions, use A⁻¹ = 1/(ad − bc) times the matrix with rows (d, −b) and (−c, a), or state No inverse.

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Test the determinant before using the formula

For a matrix with rows (a, b) and (c, d), first calculate det(A) = ad − bc. If that value is zero, the rows or columns do not provide two independent directions, so the matrix has no inverse.

When ad − bc is nonzero, form the adjugate by swapping a and d and negating b and c. Then multiply every entry by 1/(ad − bc). Keep fractional results exact and simplify them; taking four entrywise reciprocals does not produce a matrix inverse.

Learning objective

What this activity builds

Use the determinant to test invertibility, then apply the exact 2 × 2 inverse formula when the determinant is nonzero.

Worked example

Find the inverse of the matrix with rows (2, 1) and (5, 3).

The inverse has rows (3, −1) and (−5, 2).

The determinant is 2·3 − 1·5 = 1. Swap the diagonal entries, negate the off-diagonal entries and multiply by 1/1.

Teaching notes

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  • Calculate ad − bc before attempting the inverse formula.
  • When the determinant is zero, stop and state that no inverse exists.
  • Swap a and d, but negate b and c.
  • Keep every answer exact; simplify each fractional entry instead of using decimals.

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Choose whether generated matrices are always invertible, never invertible or mixed. The preview and PDF keep six existence checks followed by six exact inverse calculations.

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Math SheetNameScore/ 12TeacherDateMatrix Inverses1-6: State whether an inverse exists. 7-12: Find the inverse, or state that none exists.1. Does an inverse exist?A =30-10Yes / No2. Does an inverse exist?A =-20-46Yes / No3. Does an inverse exist?A =-2000Yes / No4. Does an inverse exist?A =-123-164Yes / No5. Does an inverse exist?A =0-40-3Yes / No6. Does an inverse exist?A =3-32-6Yes / No7. Find the inverse.A =16008. Find the inverse.A =-30-202416math-sheet.comPage 1 / 2
Math SheetNameScore/ 12TeacherDateMatrix Inverses1-6: State whether an inverse exists. 7-12: Find the inverse, or state that none exists.9. Find the inverse.A =536510. Find the inverse.A =-65-3311. Find the inverse.A =-26-6-512. Find the inverse.A =-3-26-5math-sheet.comPage 2 / 2

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