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Determinant Word Problems


There are some tricks you can do with determinants, and these tend to be turned into word problems.

  • A triangle has vertices as (1, 2), (3, 4), and (2, 3). Find the area of the triangle.
  • I could try to work from a drawing of the triangle, but this can get very complicated. Instead, I put the vertices of the triangle into a determinant, with the x-values being the first column, the corresponding y-values being the second column, and the third column all filled with 1's, like this:

      D = || 1 2 1 || 3 –4 1 || –2 3 1 ||

    Remember that the area formula for a triangle with base b and height h is A = bh. So I need the here, too. Evaluating:   Copyright Elizabeth Stapel 2004-2011 All Rights Reserved

    (1/2) D = –8

    Whoa! A negative area? Is that possible? No. So is the formula wrong? Sort of. Remember that a determinant can be negative, while areas (and absolute values) cannot. To account for this, the formula is actually "  D", where I chose the "+" or the "" so that I get a positive answer. Then:

     (1/2) D = –(1/2)(–16) = 8

      Then the area of the triangle is 8 square units.

Does it matter in which order you list the points in the determinant? No:

    another ordering

    another ordering

The order of the points in the determinant doesn't matter because changing the order of the rows in a determinant doesn't change its value, except maybe for changing the sign.

  • A parallelogram has vertices at (3, 4), (1, 2), (1, 3), and (3, 3). Find its area.
  • A triangle is half of a parallelogram, so the area of a triangle is half that of a parallelogram. Then the formula for the area of a parallelogram is just the formula for the area of a triangle, but without the "". But which three points do I plug into the formula? I've got three to choose from, and it turns out that it really doesn't matter which three I choose:

      using the points (3, 4), (1, 2) and (1, 3):

       D = 22

      using the points (3, 4), (1, 3) and (3, 3):

       D = 22

      using the points (1, 2), (1, 3) and (3, 3):

       D = 22

      using the points (3, 4), (1, 2) and (3, 3):

       D = 22

    Then the area of the parallelogram is 22 square units.

Note that, in the second and third cases, I had to change the sign to get a positive area. But in all cases, I got the same-sized value.

  • A line passes through the point (2, 3) and (3, 1). Find its equation.
  • Normally, I would find the slope, pick one of the points, and plug the slope and the point's coordinates into the slope-intercept form to find the line's equation. But I can use a determinant to find the same thing. I set up a determinant with an x-column, and y-column, and an all-ones column, and set it equal to zero, like this:

      || x y 1 || 2 3 1 || –3 1 1 || = 0

    Then I evaluate and simplify:

      (x)(3)(1) + (y)(1)(3) + (1)(2)(1) (3)(3)(1) (1)(1)(x) (1)(2)(y) = 0
      3x 3y + 2 + 9 x 2y = 0

      2x 5y + 11 = 0

      The line has equation 2x 5y = 11.

(The format for the answer to this type of exercise will vary from text to text, so make a note of how your book does it, and match that format.)

The last exercise above shows why you will need to be able to evaluate determinants by hand: your teacher can give you an exercise like this that you can't solve with your calculator. So make sure you practice enough to learn the process.

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Cite this article as:

Stapel, Elizabeth. "Determinant Word Problems." Purplemath. Available from
    http://www.purplemath.com/modules/detprobs.htm. Accessed
 

 



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