The Inverse of a Matrix and When It Exists · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Undoing a matrix with its inverse

Mathematics · Linear Algebra · ages 18-19
Name ______________________   Date ____________

  1. What is the determinant of the matrix [[1, 2], [3, 4]]?

    Answer: ______________

  2. What is the inverse of the matrix with rows (2, 0) and (0, 2)?

    • Rows (0.5, 0) and (0, 0.5)
    • Rows (2, 0) and (0, 2)
    • Rows (1, 0) and (0, 1)

  3. Only square matrices with a full set of pivots can have an inverse.

    Circle one:   True   False

  4. Which of these matrices is NOT invertible?

    • Rows (1, 2) and (3, 4)
    • Rows (2, 0) and (0, 2)
    • Rows (1, 2) and (2, 4)

  5. Which condition guarantees a square matrix has an inverse?

    • Its determinant is nonzero
    • It has a row of zeros
    • It is not square
    • Two of its rows are equal

  6. You must solve one large system Ax equals b with a single right hand side. What is the better method?

    • Compute the full inverse and multiply
    • Guess values until one fits
    • A direct elimination on the system

  7. What is the determinant of the matrix with rows (1, 2) and (3, 4)?

    Answer: ______________

  8. What is the inverse of [[2, 0], [0, 2]]?

    • [[2, 0], [0, 2]]
    • [[0.5, 0], [0, 0.5]]
    • [[0, 2], [2, 0]]
    • [[-0.5, 0], [0, -0.5]]

  9. The inverse of the matrix with rows (1, 2) and (0, 1) holds some entry in row 1, column 2. What is it?

    Answer: ______________

  10. The inverse of [[1, 2], [0, 1]] has some entry in row 1, column 2. What is it?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Undoing a matrix with its inverse W1-mt_1K6lcfMQ_b-s1

  1. -2 · Determinant is 1 times 4 minus 2 times 3 = minus 2.
  2. Rows (0.5, 0) and (0, 0.5) · Each diagonal 2 needs its reciprocal 0.5, and the zeros stay put.
  3. True · That is the entry test: anything short collapses space and cannot be undone.
  4. Rows (1, 2) and (2, 4) · Only one pair is dependent: rows (1, 2) and (2, 4), where the lower row is twice the upper, so the determinant is 0.
  5. Its determinant is nonzero · Nonzero determinant is equivalent to full pivot count, hence invertibility.
  6. A direct elimination on the system · Direct elimination solves the one system in one pass, while a full inverse adds work on top of that plus extra rounding error.
  7. -2 · Use a times d minus b times c: 1 times 4 minus 2 times 3 is minus 2.
  8. [[0.5, 0], [0, 0.5]] · Scaling by 2 undoes as scaling by 0.5, so the inverse is 0.5 times the identity.
  9. -2 · Clearing above the pivot turns row 1 into (1, minus 2) beside the identity.
  10. -2 · The inverse of [[1, 2], [0, 1]] is [[1, minus 2], [0, 1]], so the entry is minus 2.
Worksheet · LightMySky