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Given 𝐴 multiplied by 10, two, 10, one equals one, zero, zero, one, find the matrix 𝐴.
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Let’s have a look at what we’ve actually been given here.
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We’ve been given a matrix equation.
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We know that the product of 𝐴 with our matrix 10, two, 10, one is one, zero, zero, one.
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That’s the identity matrix.
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We can solve this equation a little like we would solve a normal linear equation.
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We perform inverse operations.
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Here, since we’re trying to solve for 𝐴, we’re going to multiply both sides of our equation by the inverse of 10, two, 10, one.
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Now, I’ve called 10, two, 10, one 𝐵.
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So we’re going to multiply both sides of this equation by the inverse of 𝐵.
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However, 𝐵 multiplied by the inverse of 𝐵 is the identity matrix.
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So on the left-hand side, we’ll simply be left with 𝐴.
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And the identity matrix multiplied by the inverse of 𝐵 is just the inverse of 𝐵.
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So we can see that, for our equation, 𝐴 is equal to the inverse of 𝐵.
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So we just need to find that inverse.
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𝐵 is a two-by-two matrix.
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Let’s call it in its general form 𝑎, 𝑏, 𝑐, 𝑑.
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Its inverse is one over the determinant of 𝐵 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎.
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Essentially, we swap the elements on the top left and bottom right.
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And we change the sign of the elements on the top right and bottom left.
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But what about the determinant of 𝐵?
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The determinant of 𝐵 is the product of 𝑎 and 𝑑 minus the product of 𝑏 and 𝑐.
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That’s the product of the elements on the top left and bottom right minus the product of the elements on the top right and bottom left.
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So we’ll begin then by finding the determinant of the matrix 10, two, 10, one.
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We start by multiplying the elements in the top left and the bottom right.
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That’s 10 times one.
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And we subtract the product of the elements in the top right and bottom left.
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That’s two times 10.
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10 minus 20 is negative 10.
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So the determinant of the matrix that we’ve called 𝐵 is negative 10.
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So the first part of the inverse of 𝐵 is one over the determinant of 𝐵.
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It’s one over negative 10 which is negative one-tenth.
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We’re then going to switch the elements in the top left and bottom right.
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And we’re going to change the signs of the elements in the top right and bottom left.
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So the inverse of 𝐵 is negative one-tenth multiplied by one, negative two, negative 10, 10.
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All that’s left is to multiply each element inside our matrix by negative one-tenth.
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And when we do, we see that the inverse of 𝐵 is negative one-tenth, one-fifth, one, and negative one.
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And since we said earlier that 𝐴 is going to be equal to the inverse of 𝐵, then 𝐴 is this matrix.
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It’s negative a tenth, one-fifth, one, negative one.