Question Video: Using Elementary Row Operations to Find the Inverse of Two-by-Two Matrices

Using the elementary row operation, find 𝐴⁻¹ for the matrix 𝐴 = 5, 3 and 2, 1 if possible.

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Video Transcript

Using the elementary row operation, find the inverse of 𝐴 for the matrix 𝐴 is equal to five, three, two, one if possible.

For a two-by-two matrix, 𝐴 is equal to π‘Ž, 𝑏, 𝑐, 𝑑, the inverse of 𝐴 is given by the formula one over the determinant of π‘Ž multiplied by 𝑑, negative 𝑏, negative 𝑐, π‘Ž, where the determinant of π‘Ž is given by π‘Ž multiplied by 𝑑 minus 𝑏 multiplied by 𝑐.

Notice that this means if the determinant of the matrix 𝐴 is zero, the inverse cannot exist. That’s because one over the determinant of 𝐴 would be one over zero, which we know to be undefined. Let’s substitute the values for our matrix 𝐴 into our formula first for the determinant of 𝐴.

π‘Ž multiplied by 𝑑 is five multiplied by one, and 𝑏 multiplied by 𝑐 is three multiplied by two. Evaluating, we get five minus six, which is negative one. Since the determinant of 𝐴 is not zero, the inverse of 𝐴 does indeed exist. Now that we have the determinant, we can substitute everything we know about our matrix into the formula for the inverse of 𝐴.

That’s one divided by negative one multiplied by one, negative three, negative two, five. Remember, we switch the values of the five and the one, and we make three and two negative. One divided by negative one is just negative one.

We can next multiply each of the individual elements in the matrix by negative one. And doing so, we can see that the inverse of 𝐴 is equal to negative one, three, two, negative five.

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