Video: Finding the Inverse of a Matrix

Find the multiplicative inverse of (sin πœƒ, βˆ’cos πœƒ and cos πœƒ, sin πœƒ).

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Video Transcript

Find the multiplicative inverse of sin πœƒ, negative cos πœƒ, cos πœƒ, sin πœƒ.

Remember, for a two-by-two matrix 𝐴 with elements π‘Ž, 𝑏, 𝑐, 𝑑, its inverse is given by one over the determinant of 𝐴 multiplied by 𝑑, negative 𝑏, negative 𝑐, π‘Ž, where its determinant can be found by multiplying the elements in the top left and the bottom right and then subtracting the product of the elements on the top right and the bottom left.

Notice this means if the determinant is zero, the inverse cannot exist since one over the determinant of 𝐴 will be one over zero. And we know that to be undefined.

Let’s begin by finding the determinant of the given matrix. We begin by finding the product of the top left and bottom right elements. That’s sin πœƒ multiplied by sin πœƒ. We then subtract the product of the top right and bottom left elements. That’s negative cos πœƒ multiplied by cos πœƒ. Sin πœƒ multiplied by sin πœƒ can be simply written as sin squared πœƒ. And then, we subtract negative cos squared πœƒ, which becomes plus cos squared πœƒ.

Recall the trigonometric identity. Sin squared πœƒ plus cos squared πœƒ is equal to one. And we can replace sin squared πœƒ plus cos squared πœƒ with one. And the determinant of this matrix is simply one.

So let’s substitute what we know into the formula for the inverse of the matrix. One over the determinant is one over one. We then swap the elements in the top left and the bottom right. Sin πœƒ swaps with sin πœƒ. So these actually remain unchanged. We multiply the elements on the top right and bottom left by negative one, essentially changing their sin. And we get cos πœƒ on the top right and negative cos πœƒ on the bottom left. One over one is simply one.

So the multiplicative inverse of this matrix is sin πœƒ, cos πœƒ, negative cos πœƒ, sin πœƒ.

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