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Evaluate the shown determinant where π squared is equal to negative one.
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Letβs start by using the instruction that π squared is equal to negative one and replacing every instance of π squared in our matrix with negative one.
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To find the determinant of the three-by-three matrix shown, we multiply each of the elements of the top row by the determinant of the two-by-two matrix that is not in their row or column.
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This is called their cofactor.
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Letβs start by evaluating the first cofactor.
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To find the determinant of a two-by- two matrix, we multiply the top left and bottom right elements and subtract the product of the top right and bottom left.
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Negative one multiplied by negative one minus zero multiplied by negative π is one.
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The first part of our three-by-three determinant is negative one multiplied by one.
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We now need to work out the determinant of the second two- by-two matrix.
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One plus π multiplied by negative one minus zero multiplied by one minus π is negative one minus π.
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The second part in finding the determinant for the three-by-three matrix is therefore zero multiplied by negative one minus π.
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Finally, we need to find the determinant of the third two-by-two matrix.
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Once again we find the product of the top left and bottom right and subtract the product of the top right and bottom left.
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That gives us negative π minus π squared plus one minus π, which becomes negative two π minus negative one plus one, which simplifies to negative two π plus two.
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Substituting this back into our formula for the determinant gives us one plus π multiplied by negative two π plus two.
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The first expression simplifies to negative one, and the second part of the expression simplifies to zero.
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Expanding these two brackets gives us negative one plus negative two π plus two minus two π squared plus two π.
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And then we can simplify further to give us one minus two π squared.
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Remember, π squared is negative one.
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So our expression becomes one minus two times negative one, which is three.
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The shown determinant has a value of three.