ShmoopTube

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Inverse matrices a'la Shmoop!

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After an unfortunate drugstore accident there's a hole in the space-time continuum…[Man dropping a substance and creating a green hole in the drug store]

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…Joel, the pharmacist’s assistant, finds himself inside a matrix.

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Not THE Matrix; just a matrix. [Joel trapped inside the matrix]

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Specifically, a two by two matrix with the values 1, 3, -2, and, -1.

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Suddenly… JOEL, IF YOU STAY

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HERE TOO LONG…

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…THE HOLE IN THE SPACE-TIME CONTINUUM WILL CLOSE, TRAPPING YOU HERE FOREVER. [Man wearing sunglasses ordering Joel to leave the space time continuum]

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THE ONLY WAY TO ESCAPE THIS MATRIX IS TO INVERT IT.

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ONCE THE MATRIX IS INVERTED, YOU WILL FALL BACK INTO YOUR OWN DIMENSION.

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How does Joel invert this matrix so he can get back to the right dimension? [Question on chalkboard asking Joel how can get back to his own dimension]

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First he tries turning the matrix upside down.

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That doesn’t work.

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Next he tries reversing the order of the numbers. [Joel re-ordering the numbers in the equation]

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Nope.

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Let’s help him out with some basics.

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There’s a three-step trick to finding the inverse of a two by two matrix.

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First, we find the determinant, which we can do with this formula: a times d minus b times [A finger finding the determinant of the matrices]

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c.

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We can see that a equals 1, b equals 3, c equals negative 2, and d equals negative 1…

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…so plugging this into our determinant formula, we see that 1 times negative 1 minus 3 times [numbers plugged into the determinant formula]

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negative 2… gives us negative 1 plus 6… or 5.

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The next step in finding the inverse is to switch the positions of a and d…[A and B are switched in the formula]

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…and switch the signs of c and b.

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So if we look back to our matrix, a and d switch places...

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…so the top left corner of the matrix becomes negative 1 and the bottom right corner becomes

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1.

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Then we switch the signs of c and b, so 3 becomes negative 3 and negative 2 becomes [C and B signs switched]

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positive 2.

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For the final step, take that mixed-up matrix and multiply the whole thing by 1 over the [A step of a staircase of the final step in locating the determinant]

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determinant, which is 5.

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though we get -1/5, -3/5, 2/5, and 1/5.

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Inverse matrix achieved.

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BACK TO YOUR OWN DIMENSION, PHARMACIST. Off you go [Man ordering Joel goes back to his dimension and Joel appears back in the pharmacy]

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