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. Row reduce [3 0 | 1 0; 0 5 | 0 1] until the left half is the identity. What is the entry in row 2, column 2 of the inverse? Enter it as a decimal.

    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, meaning a matrix B with AB = I and BA = I.

    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. A 2 by 3 matrix has two independent rows. Does it have an inverse?

    • Yes, because its two rows are independent
    • Yes, as long as neither row is all zeros
    • No: it is not square

  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. Row reduce the matrix with rows (1, 0) and (4, 1) beside the identity. What is its inverse?

    • Rows (1, 0) and (-4, 1)
    • Rows (1, 0) and (4, 1)
    • Rows (1, -4) and (0, 1)
    • Rows (-1, 0) and (-4, -1)

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

    Answer: ______________

  10. A has rows (2, 1) and (1, 1), and its inverse has rows (1, -1) and (-1, 2). Use the inverse to solve Ax = b for b = (4, 3). What is the first entry of x?

    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. 0.2 · Scale row 1 by 1/3 and row 2 by 1/5. The right half becomes rows (1/3, 0) and (0, 0.2), so the entry is 0.2.
  2. Rows (0.5, 0) and (0, 0.5) · Each diagonal 2 needs its reciprocal 0.5, and the zeros stay put.
  3. True · An inverse must work on both sides, AB = I and BA = I, which needs a square matrix. A missing pivot means the matrix collapses space, and that cannot be undone, so the statement is true.
  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. No: it is not square · Only square matrices can have an inverse. A 2 by 3 matrix is not square, so it has none, whatever its rows look like.
  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. Rows (1, 0) and (-4, 1) · Use R2 <- R2 - 4R1. The left half becomes the identity, and the right half becomes rows (1, 0) and (-4, 1).
  9. -2 · Clearing above the pivot turns row 1 into (1, minus 2) beside the identity.
  10. 1 · x is the inverse times b. Row 1 of the inverse with b gives 1(4) + (-1)(3) = 1, and row 2 gives -4 + 6 = 2, so x = (1, 2).
Worksheet · LightMySky